Dynamics of Polynomial Diffeomorphisms of $\mathbb {C}^2$: Combinatorial and Topological Aspects

Abstract

The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Hénon map $f_{c, b}(x, y)=(x^2+c-by, x)$ defined on $\mathbb {C}^2$ together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of $\mathbb {C}^2$ including complex Hénon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.

1 Dynamics of Polynomial Maps in One Complex Variable

Dynamics of complex Hénon maps or, more generally, polynomial diffeomorphisms of $\mathbb {C}^2$ has been a growing subject in the last 25 years.¹ The purpose of this survey paper is to discuss some results, questions and problems on this subject with an emphasis on the combinatorial and topological aspects of their Julia sets (see, e.g., Fig. 1). We regret not to touch the ergodic properties of polynomial diffeomorphisms of $\mathbb {C}^2$ (Bedford and Smillie 1992; Bedford et al. 1993a, b) and some of the recent remarkable progress such as the structural stability (Dujardin and Lyubich 2015), the classification of Fatou components (Lyubich and Peters 2014), etc. For other related topics, we recommend the survey papers (Bedford 1990, 2010, 2015; Smillie 2002).

Fig. 1

The Phoenix (Ushiki 2012). It is interesting to contrast this figure with the earlier picture (which looks like a phoenix) published in Ushiki (1988)

In this section we present ten topics on the dynamics of polynomial maps in dimension one. These topics are chosen to foreshadow the problems we will present in dimension two. For some of them we restrict our attention to the quadratic family $p_c(z)=z^2+c$. Most results in this section are well-known except for the item (vi) below where we present a new construction of automata called tight automata (Ishii and Smillie 2017) which describe the combinatorics of Julia sets.

The basic terminologies and results which appear in this section can be found in Milnor (2006). Below we use the notations $\mathbb {N}_0\equiv \mathbb {N}\cup \{0\}$, $\mathbb {T}\equiv \mathbb {R}/\mathbb {Z}$ and $\Delta \equiv \{z\in \mathbb {C} : |z|< 1\}$.

(i) Connectivity of $J_p$. Let $p : \mathbb {C}\rightarrow \mathbb {C}$ be a polynomial of degree $d\ge 2$. We call

$$\begin{aligned} K_p\equiv \big \{z\in \mathbb {C} : \{p^n(z)\}_{n\ge 0} \text{ is } \text{ bounded } \text{ in } \mathbb {C}\big \} \end{aligned}$$

the filled Julia set of p and its boundary $J_p\equiv \partial K_p$ the Julia set of p. Let

$$\begin{aligned} \mathrm {Crit}(p)\equiv \big \{z\in \mathbb {C} : p'(z)=0\big \} \end{aligned}$$

be the set of critical points of p. The following is classical.

Theorem 1.1

The Julia set $J_p$ is connected iff $\mathrm {Crit}(p)\subset K_p$.

As in Theorem 1.1 (and as we will see below), critical points play a dynamically important role. However, polynomial diffeomorphisms of $\mathbb {C}^2$ do not have critical points in the standard sense. In Sect. 3.1 we will introduce the Green function and use it to define “dynamical critical points” for such maps. To motivated it, let us first introduce the Green function for p:

$$\begin{aligned} G_p(z)\equiv \lim _{n\rightarrow \infty }\frac{1}{d^n}\log ^+|p^n(z)|, \end{aligned}$$

where $\log ^+t\equiv \max \{0, \log t\}$. One can see that $G_p$ is continuous, subharmonic and satisfies $G_p(p(z))=d\cdot G_p(z)$ on $\mathbb {C}$. It is harmonic on $\mathbb {C}{\setminus }K_p$, and $G_p(z)>0$ iff $z\in \mathbb {C}{\setminus } K_p$. Let

$$\begin{aligned} \mathrm {Crit}(G_p)\equiv \big \{z\in \mathbb {C}{\setminus } K_p : z \text{ is } \text{ a } \text{ critical } \text{ points } \text{ of } G_p \big \}. \end{aligned}$$

Since $z\in \mathrm {Crit}(G_p)$ iff $p^k(z)\in \mathrm {Crit}(p){\setminus } K_p$ for some $k\ge 0$, Theorem 1.1 yields

Corollary 1.2

The Julia set $J_p$ is connected iff $\mathrm {Crit}(G_p)=\emptyset$.

This statement will be rephrased in the context of polynomial diffeomorphisms of $\mathbb {C}^2$ in Sect. 3.1 (see Corollary 3.4) which is a theoretical basis for a computer algorithm to draw the connectedness locus in the parameter space of the complex Hénon family.

(ii) External rays for $J_p$. By Böttcher’s theorem there exists $R>0$ so that

$$\begin{aligned} \varphi _p(z)\equiv \lim _{n\rightarrow \infty }(p^n(z))^{\frac{1}{d^n}} \end{aligned}$$

defines a holomorphic map with $\varphi _p(z)/z\rightarrow 1$ as $|z|\rightarrow \infty$ (by choosing an appropriate $d^n$-th root) and satisfies $\varphi _p(p(z))=(\varphi _p(z))^d$ for $|z|>R$, which serves as the Böttcher coordinate of p near $\infty$. We also have $G_p(z)=\log |\varphi _p(z)|$ for $|z|>R$.

Now assume that $J_p$ is connected. Then, $(\mathbb {C}\cup \{\infty \}){\setminus } K_p$ is a simply connected domain in the Riemann sphere. Therefore,

Theorem 1.3

If $J_p$ is connected, then the map $\varphi _p$ extends to a conformal isomorphism:

$$\begin{aligned} \varphi _p : \mathbb {C}{\setminus } K_p\longrightarrow \mathbb {C}{\setminus } \overline{\Delta } \end{aligned}$$

which satisfies $(\varphi _p(z))^d=\varphi _p(p(z))$.

Definition 1.4

We call $R_p(\theta )\equiv \{\varphi _p^{-1}(re^{2\pi i\theta }) : r>1\}$ the external ray of angle $\theta \in \mathbb {T}$ for $K_p$.

An external ray $R_p(\theta )$ is said to land on a point $z_p(\theta )\in J_p$ if the limit point $\lim _{r\downarrow 1}\varphi _p^{-1}(re^{2\pi i\theta })$ exists and is equal to $z_p(\theta )$.

(iii) Expansion on $J_p$. Recall the following notion.

Definition 1.5

We say that a polynomial map p is expanding on $J_p$ if there exist $C>0$ and $\lambda >1$ so that for any $z\in J_p$ we have $\Vert (p^n)'(z)\Vert \ge C\lambda ^n$ ($n\ge 0$), where $\Vert \cdot \Vert$ is the norm with respect to the spherical metric.

The next classical result provides a criterion for a polynomial p to be expanding.

Theorem 1.6

A polynomial map p is expanding on $J_p$ iff $p^n(c)$ either converges to an attractive cycle of p or tends to infinity for every $c\in \mathrm {Crit}(p)$.

The proof is supplied by using the Poincaré metric defined in a neighborhood of $J_p$.

(iv) Quotient of a circle. Assume that $J_p$ is connected and p is expanding on $J_p$. Then, $R_p(\theta )$ is shown to land on a point $z_p(\theta )\in J_p$ for any $\theta \in \mathbb {T}$. This gives rise to a continuous surjection:

$$\begin{aligned} \psi _p : \mathbb {T}\ni \theta \longmapsto z_p(\theta )\in J_p \end{aligned}$$

which satisfies $\psi _p(\delta _d(\theta ))=p(\psi _p(\theta ))$, where $\delta _d(\theta )\equiv d\cdot \theta$. For $\theta , \theta '\in \mathbb {T}$, we write $\theta \sim _{\mathrm {DH}}\theta '$ iff $\psi _p(\theta )=\psi _p(\theta ')$. Then the quotient dynamics $\delta _d/_{\sim _{\mathrm {DH}}} : \mathbb {T}/_{\sim _{\mathrm {DH}}}\rightarrow \mathbb {T}/_{\sim _{\mathrm {DH}}}$ is well-defined.

Theorem 1.7

Assume that $J_p$ is connected and p is expanding on $J_p$. Then, the factor map $\psi _p/_{\sim _{\mathrm {DH}}} : \mathbb {T}/_{\sim _{\mathrm {DH}}}\rightarrow J_p$ is a topological conjugacy from $\delta _d/_{\sim _{\mathrm {DH}}} : \mathbb {T}/_{\sim _{\mathrm {DH}}}\rightarrow \mathbb {T}/_{\sim _{\mathrm {DH}}}$ to $p : J_p\rightarrow J_p$.

The equivalence relation $\sim _{\mathrm {DH}}$ is well understood (see Douady 1993; Thurston 2009 and (v), (vi), (vii) and (ix) below).

(v) Hubbard trees. Hubbard trees in $\mathbb {C}$ have been originally defined by J. H. Hubbard in Orsay Notes (Douady and Hubbard 1985) (see also Douady 1993). Here we introduce Hubbard trees presented in Ishii (2009) which is given in the framework of multivalued dynamical systems (Ishii and Smillie 2010) (the original definition of a Hubbard tree is a single space $\mathcal {T}^0$ defined below). Let us first recall some terminologies from Ishii and Smillie (2010).

A multivalued dynamical system is a quadruple $(X^0, X^1; \iota _X, f)$ where $X^0$ and $X^1$ are a pair of spaces and $\iota _X, f : X^1\rightarrow X^0$ is a pair of maps between them. Note that $\iota _X$ is not necessarily injective. If $\iota _X$ is injective, then $\iota _X^{-1}\circ f$ is single-valued, and if $\iota _X$ is not injective, then $\iota _X^{-1}\circ f$ is multivalued. The standard construction of the pullbacks of spaces gives a sequence of multivalued dynamical systems $\iota _X, f : X^{n+1}\rightarrow X^n$, where $X^n$ is the space of orbits of length n:

$$\begin{aligned} X^n=\big \{(x_0, \dots , x_{n-1})\in (X^1)^n : f(x_k)=\iota _X(x_{k+1})\big \}. \end{aligned}$$

This gives rise to the one-sided orbit space:

$$\begin{aligned} X^{+\infty }\equiv \big \{(x_k)_{k\ge 0}\in (X^1)^{\mathbb {N}_0} : f(x_k)=\iota _X(x_{k+1}) \big \} \end{aligned}$$

and the two-sided orbit space:

$$\begin{aligned} X^{\pm \infty }\equiv \big \{(x_k)_{k\in \mathbb {Z}}\in (X^1)^{\mathbb {Z}} : f(x_k)=\iota _X(x_{k+1}) \big \} \end{aligned}$$

as well as the shift maps $f^{+\infty } : X^{+\infty }\rightarrow X^{+\infty }$ and $f^{\pm \infty } : X^{\pm \infty }\rightarrow X^{\pm \infty }$ respectively.

A classical dynamical system $f : X\rightarrow X$ can be interpreted as a multivalued dynamical system by letting $X^0=X^1\equiv X$ and $\iota _X : X^1\rightarrow X^0$ to be the identity map. In this case, $X^{\pm \infty }$ can be identified with the so-called natural extension of $f : X\rightarrow X$. A polynomial-like map $f : U\rightarrow V$ with $\overline{U}\subset V$ is regarded as a multivalued dynamical system by letting $X^0\equiv V$, $X^1\equiv U$ and $\iota _X : U\rightarrow V$ is the inclusion map (a similar idea has been introduced in Kahn (2006) for the study of renormalization of polynomial maps). When both $X^0$ and $X^1$ are finite sets, any pair of maps $\iota _X, f : X^1\rightarrow X^0$ can be interpreted as a finite directed graph; the vertex set is $X^0$ and we regard an element $x\in X^1$ as an arrow from $\iota _X(x)\in X^0$ to $f(x)\in X^0$. In this case, the orbit space $X^{+\infty }$ (resp. $X^{\pm \infty }$) is a one-sided (resp. two-sided) subshift of finite type.

An important class of multivalued dynamical systems is

Definition 1.8

Let $(X^m, d^m)$ ($m=0, 1$) be complete length spaces. A multivalued dynamical system $\iota _X, f : X^1\rightarrow X^0$ is called expanding if (i) $f : X^1\rightarrow X^0$ is a covering, and (ii) there exist $\delta >0$ and $\lambda >1$ so that $d^0(f(x), f(x'))\ge \lambda \cdot d^0(\iota _X(x), \iota _X(x'))$ holds whenever $d^1(x, x')<\delta$.

Now we formulate a Hubbard tree as a multivalued dynamical system after Ishii (2009) based on the following notions (Douady 1993). Throughout this section we assume that any critical point of p is either periodic or tends to infinity. Let $A^0$ be the set of superattractive periodic points of p and set $A^1\equiv p^{-1}(A^0)$. For each connected component U of $\mathrm {int}(K_p)$ there is a unique $a\in U$ which is eventually mapped to $A^0$. Let $\chi _U : U\rightarrow \Delta$ be a Böttcher coordinate of U so that $\chi _U(a)=0$. Since $\partial U$ is locally connected, this extends to a homeomorphism $\chi _U : \overline{U}\rightarrow \overline{\Delta }$. An internal ray in U is the inverse image of a ray in $\Delta$ by $\chi _U$. An arc $\gamma \subset K_p$ is called a legal arc if for any connected component U of $\mathrm {int}(K_p)$, the intersection $\gamma \cap U$ is contained in the union of two rays in U. Then, any two points in $K_p$ is connected by a unique legal arc. The legal hull of a finite subset of $K_p$ is the union of such legal arcs connecting any two points in the finite subset.

For $m=0, 1$, the vein $\mathcal {H}^m$ is defined as the legal hull of $A^m$ in the filled Julia set $K_p$. If a point $a\in \mathcal {H}^m$ belongs to $A^m$, we replace $a\in \mathcal {H}^m$ by a loop to obtain a tree decorated with loops denoted by $\mathcal {T}^m$. The polynomial map p naturally induces a map $\tau : \mathcal {T}^1\rightarrow \mathcal {T}^0$ up to homotopy. One can also define a continuous map $\iota _{\mathcal {T}} : \mathcal {T}^1\rightarrow \mathcal {T}^0$ up to homotopy which is the identity on $\mathcal {T}^0$ and smashes each connected component of $\mathcal {T}^1{\setminus }\mathcal {T}^0$ to a point in $\mathcal {T}^0$.

Definition 1.9

We call the multivalued dynamical system $\iota _{\mathcal {T}}, \tau : \mathcal {T}^1\rightarrow \mathcal {T}^0$ the Hubbard tree.

Let $\tau ^{+\infty } : \mathcal {T}^{+\infty }\rightarrow \mathcal {T}^{+\infty }$ be the shift map on the one-sided orbit space of $\iota _{\mathcal {T}}, \tau : \mathcal {T}^1\rightarrow \mathcal {T}^0$.

Theorem 1.10

Assume that any critical point of of p is either periodic or tends to infinity. Then, $p : J_p\rightarrow J_p$ is topologically conjugate to $\tau ^{+\infty } : \mathcal {T}^{+\infty }\rightarrow \mathcal {T}^{+\infty }$.

A proof can be found in Ishii (2009) which uses the idea of homotopy shadowing developed in Ishii and Smillie (2010).

(vi) Tight automata. In his PhD thesis, Oliva (1998) has given a recipe to construct automata which describe the equivalence relation $\sim _{\mathrm {DH}}$ in Theorem 1.7 for some real quadratic polynomials. This recipe was supported by a great deal of evidence but without a formal proof. Following our forthcoming paper (Ishii and Smillie 2017) we here construct an automaton called a tight automaton for any expanding polynomial map and justify the argument of Oliva. In Ishii and Smillie (2017) we explain the construction only for the quadratic case, but here we discuss a polynomial of arbitrary degree $d\ge 2$. Throughout the item (vi) we assume that any critical point of p is either periodic or tends to infinity. In particular, p is expanding on $J_p$.

Remark 1.11

In Ishii and Smillie (2017) we will introduce a yet another version of Hubbard tree called a homotopy Hubbard tree as a purely homotopical object. We will demonstrate that this notion not only fits better to the construction of tight automata but also unifies several other combinatorial descriptions of Julia sets such as Thurston’s lamination theory (Thurston 2009).

Let $V^0$ be a neighborhood of $K_p$ which does not contain any critical points of p in $\mathbb {C}{\setminus } K_p$ and satisfies $\overline{p^{-1}(V^0)}\subset V^0$. Take a neighborhood $U^0$ of the set $A^0$ of superattractive periodic points of p so that $\overline{p(U^0)}\subset U^0$. Let $W^0\equiv V^0{\setminus } \overline{U^0}$ and $W^1 \equiv p^{-1}(W^0)$. This defines a multivalued dynamical system $\iota _W, p : W^1\rightarrow W^0$, where $\iota _W$ is the inclusion.

Fig. 2

Tight paths in $W^0$ (left) and in $W^1$ (right) for the Basilica map (Ishii and Smillie 2017)

Fig. 3

Tight paths in $W^0$ (left) and in $W^1$ (right) for the Rabbit map (Ishii and Smillie 2017)

Denote by $z_p(\theta )$ the landing point of the external ray $R_p(\theta )$. Let $\widehat{\mathcal {H}}^0$ be the legal hull of $\{z_p(0)\}\cup A^0$ in $K_p$ and define $\widehat{\mathcal {T}}^0\equiv \widehat{\mathcal {H}}^0\cup R_p(0)$. Similarly, let $\widehat{\mathcal {H}}^1$ be the legal hull of $\{z_p(0), z_p(\frac{1}{d}), \dots , z_p(\frac{d-1}{d})\}\cup A^1$ in $K_p$ and define $\widehat{\mathcal {T}}^1\equiv \widehat{\mathcal {H}}^1\cup R_p(0)\cup R_p(\frac{1}{d})\cup \cdots \cup R_p(\frac{d-1}{d})$. We call the pair of spaces $\widehat{\mathfrak {T}}\equiv (\widehat{\mathcal {T}}^0, \widehat{\mathcal {T}}^1)$ the extended Hubbard tree of p.

Choose a basepoint $b\in R_p(\frac{1}{2})$ and set $\{b_0, b_1\dots , b_{d-1}\}\equiv p^{-1}(b)$ so that $b_k\in R_p(\frac{2k+1}{2d})$.

Definition 1.12

A path in $W^0$ from b to itself which intersects $\widehat{\mathcal {T}}^0$ transversally at most once is called a tight path in $W^0$. A path in $W^1$ from a point in $p^{-1}(b)$ to a point in $p^{-1}(b)$ which intersects $\widehat{\mathcal {T}}^1$ transversally at most once is called a tight path in $W^1$.

See Fig. 2 where the blue curves² represent the boundaries of $W^m$, green curves represent the segment part of $\widehat{\mathcal {T}}^m$ and the red curves represent the tight paths in $W^m$ ($m=0, 1$) for the Basilica map. Figure 3 describes the corresponding objects for the Rabbit map.

The homotopy class of a path in $W^0$ from b to itself relative to endpoints is called a tight homotopy class in $W^0$ if it contains a tight path in $W^0$. The homotopy class of a path in $W^1$ from a point in $p^{-1}(b)$ to a point in $p^{-1}(b)$ relative to endpoints is called a tight homotopy class in $W^1$ if it contains a tight path in $W^1$.

Now we construct a labeled directed graph as follows. Fix a family of paths $\gamma _i$ in $W^0{\setminus } \widehat{\mathcal {T}}^0$ from b to $b_i$. The vertex set consists of all tight homotopy classes in $W^0$. The arrow set consists of all tight homotopy classes in $W^1$. When $[\gamma ]$ is the tight homotopy class of a tight path $\gamma$ from $b_i$ to $b_j$ in $W^1$, one can check that both $[p(\gamma )]$ and $[\gamma _i\cdot \iota _W(\gamma )\cdot \gamma _j^{-1}]$ are tight in $W^0$, where $\cdot$ denotes the concatenation of two paths and $\gamma ^{-1}$ is the time reversal of $\gamma$. Therefore, such $[\gamma ]$ can be regarded as an arrow from its tail $[\gamma _i\cdot \iota _W(\gamma )\cdot \gamma _j^{-1}]$ to its head $[p(\gamma )]$ and we label it as $(i, j)\in \Sigma _d^2$, where $\Sigma _d\equiv \{0, \dots , d-1\}$. This gives a directed labeled graph denoted by $\mathfrak {A}_T(\widehat{\mathfrak {T}})$.

Definition 1.13

The directed labeled graph $\mathfrak {A}_T(\widehat{\mathfrak {T}})$ is called a tight automaton of $\widehat{\mathfrak {T}}$.

Denote by $\Sigma _d^{\mathbb {N}_0}\equiv \{\varepsilon _0\varepsilon _1\cdots : \varepsilon _i\in \Sigma _d\}$ the space of all one-sided sequences over $\Sigma _d$ equipped with the product topology. Let $\sigma : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$ be the shift map given by $\sigma (\varepsilon _0\varepsilon _1\cdots )\equiv \varepsilon _1\varepsilon _2\cdots$.

Let $\mathfrak {A}_T=\mathfrak {A}_T(\widehat{\mathfrak {T}})$ be the tight automaton of $\widehat{\mathfrak {T}}$. For $\underline{\varepsilon }=(\varepsilon _n)_{n\in \mathbb {N}_0}, \underline{\varepsilon }'=(\varepsilon '_n)_{n\in \mathbb {N}_0}\in \Sigma _d^{\mathbb {N}_0}$ we write $\underline{\varepsilon }\sim _{\mathfrak {A}_T}\underline{\varepsilon }$ if there exists a sequence of successive arrows in $\mathfrak {A}_T$ along which the sequence of labelings is $(\varepsilon _n, \varepsilon _n')_{n\in \mathbb {N}_0}$. This defines the factor $\sigma /_{\sim _{\mathfrak {A}_T}} : \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_T}}\rightarrow \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_T}}$ of the shift map. The next result shows that the tight automaton describes the combinatorics of the Julia set.

Theorem 1.14

(Ishii and Smillie 2017) Let p be a polynomial of degree $d\ge 2$ and assume that any critical point of p is either periodic or tends to infinity. Then, $p : J_p\rightarrow J_p$ is topologically conjugate to $\sigma /_{\sim _{\mathfrak {A}_T}} : \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_T}}\rightarrow \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_T}}$.

Next we explain a recipe à la Oliva (1998) to compute tight automata in terms of the extended Hubbard tree alone. For $z\in \widehat{\mathcal {T}}^m$ ($m=0, 1$) the number of connected components of $\widehat{\mathcal {T}}^m{\setminus } \{z\}$ is called the valency at z and denoted by v(z). A point $z\in \widehat{\mathcal {T}}^m$ is said to be branching if $v(z)\ge 3$. Let $B^m$ the set of branching points in $\widehat{\mathcal {T}}^m$. The trunk $\mathcal {T}_{\mathrm {tr}}$ of $\widehat{\mathcal {T}}^1$ is the union of the legal hull of $\big \{z_p(0), \ldots , z_p(\frac{d-1}{d})\big \}$ in $K_p$ and $R_p(0)\cup \cdots \cup R_p(\frac{d-1}{d})$. Note that $\mathcal {T}_{\mathrm {tr}}$ cuts $W^1$ into d pieces.

Recipe for $\mathfrak {A}_E$. Consider the following multivalued dynamical system:

$$\begin{aligned} \iota _T, \tau : \widehat{\mathcal {T}}^1\cup \{b_0, \dots , b_{d-1}\}\longrightarrow \widehat{\mathcal {T}}^0\cup \{b\}, \end{aligned}$$

where $\iota _T$ is the identity map on $R_p(0)$, $\iota _T(z)\equiv b$ is the constant map on $R_p(\frac{1}{d})\cup \cdots \cup R_p(\frac{d-1}{d})$, and $\iota _T(b_k)=\tau (b_k)\equiv b$ for $k\in \Sigma _d$.

• (1) A connected component of $\widehat{\mathcal {T}}^m{\setminus } (\{\text{ loops } \text{ in } \mathcal {T}^m\}\cup B^m)$ is called a segment in $\widehat{\mathcal {T}}^m$. Denote by $S^m$ the set of segments in $\widehat{\mathcal {T}}^m$. To each segment $s\in S^m$ we associate two directions to obtain the two directed segments denoted by $s^+$ and $s^-$. Let $\widetilde{S}^m$ be the totality of the directed segments in $\widehat{\mathcal {T}}^m$. For two directed subsegments s and $s'$ in $\widehat{\mathcal {T}}^m$, we write $s\propto s'$ if $s\subset s'$ as subsets of $\widehat{\mathcal {T}}^m$ and the orientations of s and $s'$ coincide.

• (2) We let $\widetilde{S}^1\cup \{b_0, \dots , b_{d-1}\}$ be the arrow set and $\widetilde{S}^0\cup \{b\}$ be the vertex set of the directed graph as follows. Choose $a\in \widetilde{S}^1\cup \{b_0, \dots , b_{d-1}\}$.

  • When both $\iota _T(a)$ and $\tau (a)$ are directed subsegments in $\widehat{\mathcal {T}}^0$, we draw an arrow from $v\in \widetilde{S}^0$ to $v'\in \widetilde{S}^0$ so that $\iota _T(a)\propto v$ and $\tau (a)\propto v'$ are satisfied.

  • When $\iota _T(a)$ is a point and $\tau (a)$ is a directed subsegment in $\widehat{\mathcal {T}}^0$, we draw an arrow from b to $v'\in \widetilde{S}^0$ so that $\tau (a)\propto v'$ is satisfied.

  • When both $\iota _T(a)$ and $\tau (a)$ are points (this happens exactly when $a=b_k$ for some $k\in \Sigma _d$), we draw an arrow from b to itself.

  This give a new multivalued dynamical system:

  $$\begin{aligned} \iota _T, \tau : \widetilde{S}^1\cup \{b_0, \dots , b_{d-1}\}\longrightarrow \widetilde{S}^0\cup \{b\}. \end{aligned}$$

  Note that there are $2(d-1)$ arrows which represent the directed segments corresponding to $R_p(\frac{1}{d}), \dots , R_p(\frac{d-1}{d})\in S^1$ from b to the directed segments corresponding to $R_p(0)\in S^0$.

• (3) We label the arrows as follows to obtain an automaton $\mathfrak {A}_E$.

  • When $a\in \widetilde{S}^1$ is a subset of $\mathcal {T}_{\mathrm {tr}}$, we label the arrow by $(k, k')$, where $b_k$ (resp. $b_{k'}$) belongs to the right-hand (resp. left-hand) component of $W^1{\setminus } \mathcal {T}_{\mathrm {tr}}$.

  • When $a\in \widetilde{S}^1$ is a subset of $\widehat{\mathcal {T}}^1{\setminus } \mathcal {T}_{\mathrm {tr}}$, we label the arrow by (k, k), where $b_k$ belongs to the connected component of $W^1{\setminus } \mathcal {T}_{\mathrm {tr}}$ containing a.

  • When $a=b_k$, we label the arrow by (k, k).

Fig. 4

Directed segments for the Basilica map (Ishii and Smillie 2017)

Fig. 5

Directed segments for the Rabbit map (Ishii and Smillie 2017)

Fig. 6

Tight automaton for the Basilica Julia set (Ishii and Smillie 2017)

Fig. 7

Tight automaton for the Rabbit Julia set (Ishii and Smillie 2017)

(end of recipe for $\mathfrak {A}_E$)

The directed segments for the Basilica map and the Rabbit map are presented in Figs. 4 and 5 respectively. The tight automata for the Basilica map and the Rabbit map computed through the recipe above are presented in Figs. 6 and 7 respectively.

Let $M\equiv \max \{v(z) : z\in \widehat{\mathcal {T}}^1\}$. We have

Theorem 1.15

(Ishii and Smillie 2017) Assume that $M\le 3$. Then, $\mathfrak {A}_T=\mathfrak {A}_E$.

The above theorem applies to the real quadratic polynomials as well as the Rabbit map. In particular, this justifies the observation of Oliva (1998). A statement without the assumption $M\le 3$ requires an additional automaton $\mathfrak {A}_V$ and shows that the “join” $\mathfrak {A}_E\cup \mathfrak {A}_V$ is identical to $\mathfrak {A}_T$ (see Ishii and Smillie 2017 for more details). The proofs of the results in the item (vi) use the idea of homotopy shadowing (Ishii and Smillie 2010) and a “duality” between a directed segment and a tight path.

(vii) Iterated monodromy groups. In Nekrashevych (2005) Volomydir Nekrashevych has developed a group-theoretic framework to describe the dynamics of branched partial self-covering, which led the solution to the so-called twisted rabbit conjecture (Bartholdi and Nekrashevych 2006). Throughout the item (vii) we assume that $J_p$ is connected and p is expanding on $J_p$. Then, as in (vi) one can take a path-connected neighborhood $W^0$ of $J_p$ so that $\iota _W, p : W^1\rightarrow W^0$ is an expanding system, where $W^1 \equiv p^{-1}(W^0)$.

For the multivalued dynamical system $\mathfrak {W}=(W^0, W^1; \iota _W, p)$, one can define its pullbacks $\iota _W, p : W^n\rightarrow W^{n-1}$ so that the iterations $\iota _W^n, p^n : W^n\rightarrow W^0$ are well-defined. By the definition of $W^m$ we see that $p : W^1\rightarrow W^0$ is a covering of degree $d\ge 2$. Fix a base-point $b\in W^0$. We define $T^{*}\equiv \bigsqcup _{n=0}^{\infty }p^{-n}(b)$ and draw an arrow from $y\in p^{-n-1}(b)$ to $y'\in p^{-n}(b)$ whenever $p(y)=y'$. The directed rooted d-regular tree obtained in this way is called the preimage tree and denoted by T. Since $p : W^1\rightarrow W^0$ is a covering, the fundamental group $\pi _1(W^0, b)$ acts on $p^{-n}(b)$ for each $n\ge 0$, hence on T. Let the homomorphism $\phi : \pi _1(W^0, b) \rightarrow \mathfrak {S}(T)$ be the action of $\pi _1(W^0, b)$ on T. Following Nekrashevych (2005), (see also Bartholdi et al. 2003) we define

Definition 1.16

We call

$$\begin{aligned} \mathrm {IMG}(\mathfrak {W}) \equiv \pi _1(W^0, b) / \mathrm {Ker}(\phi ) \end{aligned}$$

the iterated monodromy group for the multivalued dynamical system $\mathfrak {W}=(W^0, W^1; \iota _W, p)$.

Let $\Sigma _d^n$ be the set of words of length $n\ge 0$ over $\Sigma _d$ and put $\Sigma _d^{*}\equiv \bigsqcup _{n=0}^{\infty }\Sigma _d^n$, where $\Sigma _d^0$ consists of the empty word $\emptyset$. Fix a bijection $\Lambda : \Sigma _d\rightarrow p^{-1}(b)\subset W^1$ and a family of paths $\{l_{\varepsilon }\}_{\varepsilon \in \Sigma _d}$ where $l_{\varepsilon }$ connects b to $\iota _W(\Lambda (\varepsilon ))$ in $W^0$. For $n\ge 1$ and $\varepsilon _0\cdots \varepsilon _n\in \Sigma _d^{n+1}$ we inductively define a path $l_{\varepsilon _0\cdots \varepsilon _n}$ in $W^0$ as follows. Assume that $l_{\varepsilon _1\cdots \varepsilon _n}$ is determined for any $\varepsilon _1\cdots \varepsilon _n\in \Sigma _d^n$. We put

$$\begin{aligned} l_{\varepsilon _0\varepsilon _1\cdots \varepsilon _n}\equiv l_{\varepsilon _0}\cdot \iota _W (p^{-1}(l_{\varepsilon _1\cdots \varepsilon _n})_{\Lambda (\varepsilon _0)}), \end{aligned}$$

where $p^{-1}(l_{\varepsilon _1\cdots \varepsilon _n})_{\Lambda (\varepsilon _0)}$ is the lift of $l_{\varepsilon _1\cdots \varepsilon _n}$ by p whose initial point is $\Lambda (\varepsilon _0)$.

Given a path l, let e(l) be its end point and put

$$\begin{aligned} \Lambda (\varepsilon _0\varepsilon _1\cdots \varepsilon _n)\equiv e(p^{-1}(l_{\varepsilon _1\cdots \varepsilon _n})_{\Lambda (\varepsilon _0)}). \end{aligned}$$

Since we can verify $p(\Lambda (\varepsilon _0\cdots \varepsilon _n))=\iota _W(\Lambda (\varepsilon _1\cdots \varepsilon _n))$, the finite sequence:

$$\begin{aligned} \widetilde{\Lambda }(\varepsilon _0\varepsilon _1\cdots \varepsilon _n)\equiv (\Lambda (\varepsilon _0\cdots \varepsilon _n), \Lambda (\varepsilon _1\cdots \varepsilon _n), \dots ,\Lambda (\varepsilon _n)) \end{aligned}$$

gives a point in $p^{-n-1}(b)$. This defines $\widetilde{\Lambda } : \Sigma _d^{n+1}\rightarrow p^{-n-1}(b)$, which gives rise to an isomorphism:

$$\begin{aligned} \widetilde{\Lambda } : \Sigma _d^{*}\longrightarrow T^{*}, \end{aligned}$$

where we set $\widetilde{\Lambda }(\emptyset )\equiv b$ (see Proposition 5.3 in Bartholdi et al. 2003). The action of $\mathrm {IMG}(\mathfrak {W})$ on $T^{*}$ induces an action on $\Sigma _d^{*}$ which we call the standard action of $\mathrm {IMG}(\mathfrak {W})$ on $\Sigma _d^{*}$.

Definition 1.17

We say that $\underline{\varepsilon }=(\varepsilon _n)_{n\ge 0}$ and $\underline{\varepsilon }'=(\varepsilon '_n)_{n\ge 0}$ in $\Sigma _d^{\mathbb {N}_0}$ are asymptotically equivalent and write $\underline{\varepsilon }\sim _{\mathrm {asym}} \underline{\varepsilon }'$ if there exists a finite set $F\subset \mathrm {IMG}(\mathfrak {W})$ so that one can find $\gamma _n\in F$ with

$$\begin{aligned} (\varepsilon _0\cdots \varepsilon _n)^{\gamma _n}=\varepsilon '_0\cdots \varepsilon '_n \end{aligned}$$

for any $n\ge 0$.

It is easy to see that the asymptotic equivalence forms an equivalence relation. The quotient space $\Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathrm {asym}}}$ is called the limit space of $\mathrm {IMG}(\mathfrak {W})$. The shift map $\sigma : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$ defines a factor map $\sigma /_{\sim _{\mathrm {asym}}} : \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathrm {asym}}}\rightarrow \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathrm {asym}}}$. Nekrashevych proved the following.

Theorem 1.18

(see Theorem 9.7 in Bartholdi et al. 2003) Let p be a polynomial of degree $d\ge 2$ and assume that $J_p$ is connected and p is expanding on $J_p$. Then, $p : J_p\rightarrow J_p$ is topologically conjugate to the factor map $\sigma /_{\sim _{\mathrm {asym}}} : \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathrm {asym}}}\rightarrow \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathrm {asym}}}$.

The action of $\mathrm {IMG}(\mathfrak {W})$ has the following significant property called the self-similarily.

Proposition 1.19

(Proposition 5.4 in Bartholdi et al. 2003) For every $g \in \mathrm {IMG}(\mathfrak {W})$ and $\varepsilon \in \Sigma _d$ there exist unique $g|_{\varepsilon }\in \mathrm {IMG}(\mathfrak {W})$ and $\varepsilon '\in \Sigma _d$ so that $(\varepsilon \underline{w})^g=\varepsilon '(\underline{w})^{g|_{\varepsilon }}$ holds for any word $\underline{w}\in \Sigma _d^{*}$.

This property allows us to define two maps:

$$\begin{aligned} \pi : \Sigma _d\times \mathrm {IMG}(\mathfrak {W}) \longrightarrow \mathrm {IMG}(\mathfrak {W}) \end{aligned}$$

by $\pi (\varepsilon , g)\equiv g|_{\varepsilon }$ and

$$\begin{aligned} \lambda : \Sigma _d\times \mathrm {IMG}(\mathfrak {W})\longrightarrow \Sigma _d \end{aligned}$$

by $\lambda (\varepsilon , g)\equiv \varepsilon '$. Given a word $\varepsilon _0\cdots \varepsilon _n\in \Sigma _d^{n+1}$ we inductively define $g|_{\varepsilon _0\cdots \varepsilon _n}\equiv (g|_{\varepsilon _0\cdots \varepsilon _{n-1}})|_{\varepsilon _n}$.

The nucleus of $\mathrm {IMG}(\mathfrak {W})$ is defined as

$$\begin{aligned} \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}\equiv \bigcup _{g\in G}\bigcap _{n\in \mathbb {N}}\bigcup _{|\underline{w}|\ge n} \big \{g|_{\underline{w}}\big \}, \end{aligned}$$

where $|\underline{w}|$ denotes the length of the word $\underline{w}$. It can be shown that the nucleus $\mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}$ is finite when the polynomial p is expanding on $J_p$ (see the first half of Theorem 9.7 in Bartholdi et al. 2003). Moreover, the above two maps restrict to the nucleus to obtain $\pi : \Sigma _d\times \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})} \rightarrow \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}$ and $\lambda : \Sigma _d\times \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})} \rightarrow \Sigma _d$. This gives a directed labeled graph as follows; the vertex set is $\mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}$ and we draw an arrow from $g\in \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}$ to $g'\in \mathcal {N}_{\mathrm {IMG}(\mathfrak {W})}$ iff $g'=\pi (\varepsilon , g)$ holds for some $\varepsilon \in \Sigma _d$ and label the arrow as $(\varepsilon , \lambda (\varepsilon , g))$.

Definition 1.20

The directed labeled graph obtained in this way is called the IMG automaton of $\mathfrak {W}$ and denoted by $\mathfrak {A}_{\mathrm {IMG}}(\mathfrak {W})$.

Let $\mathfrak {A}_{\mathrm {IMG}}=\mathfrak {A}_{\mathrm {IMG}}(\mathfrak {W})$ be the IMG automaton of $\mathfrak {W}$. As for a tight automaton this defines an equivalence relation $\sim _{\mathfrak {A}_{\mathrm {IMG}}}$ in $\Sigma _d^{\mathbb {N}_0}$. In Proposition 9.2 of Bartholdi et al. (2003) it was shown that the asymptotic equivalence $\sim _{\mathrm {asym}}$ and $\sim _{\mathfrak {A}_{\mathrm {IMG}}}$ are identical. Therefore, Theorem 1.18 yields

Corollary 1.21

(Nekrashevych) Let p be a polynomial of degree $d\ge 2$ and assume that $J_p$ is connected and p is expanding on $J_p$. Then, $p : J_p\rightarrow J_p$ is topologically conjugate to the factor map $\sigma /_{\sim _{\mathfrak {A}_{\mathrm {IMG}}}} : \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_{\mathrm {IMG}}}}\rightarrow \Sigma _d^{\mathbb {N}_0}/_{\sim _{\mathfrak {A}_{\mathrm {IMG}}}}$.

See Nekrashevych (2005), Bartholdi and Nekrashevych (2006), Bartholdi et al. (2015) for more details. In Ishii and Smillie (2017) we plan to discuss the relationship between the tight automata in (vi) and the IMG automata in (vii). Note that a polynomial diffeomorphism of $\mathbb {C}^2$ can not be a covering and can not be expanding in a neighborhood of the Julia set. Therefore, formulating iterated monodromy groups for such class of dynamical systems is not obvious (see Ishii 2014 for more details).

(viii) Monodromy representation on shift space. Let $\sigma : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$ be the shift map on the space of one-sided symbol sequences with d symbols. Recall that the space $\Sigma _d^{\mathbb {N}_0}$ inherits the product topology. A shift automorphism of degree d is a homeomorphism $\tau : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$ which commutes with the shift map $\sigma : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$, i.e. $\sigma \circ \tau =\tau \circ \sigma$. Denote by $\mathrm {Aut}(\Sigma _d^{\mathbb {N}_0}, \sigma )$ the space of all shift automorphisms of degree d. This space forms a group under composition.

Any polynomial of degree $d\ge 2$ is affine conjugate to a polynomial of the form $p(z)=z^d+a_{d-2}z^{d-2}+\cdots +a_0$. Below we identify p with the point $(a_{d-2}, \dots , a_0)\in \mathbb {C}^{d-1}$. The connectedness locus is defined as

$$\begin{aligned} \mathcal {M}_d\equiv \big \{ p\in \mathbb {C}^{d-1} : J_p \text{ is } \text{ connected } \big \}=\big \{ p\in \mathbb {C}^{d-1} : \mathrm {Crit}(p) \subset K_p \big \} \end{aligned}$$

and the shift locus is defined as

$$\begin{aligned} \mathcal {S}_d\equiv \big \{ p\in \mathbb {C}^{d-1} : \mathrm {Crit}(p) \cap K_p=\emptyset \big \}. \end{aligned}$$

By using the Poincaré metric one can easily show that for $p\in \mathcal {S}_d$, the restriction $p : J_p\rightarrow J_p$ is expanding and topologically conjugate to $\sigma : \Sigma _d^{\mathbb {N}_0}\rightarrow \Sigma _d^{\mathbb {N}_0}$. Note that $\mathcal {M}_2\sqcup \mathcal {S}_2=\mathbb {C}$ holds, but $\mathcal {M}_d\sqcup \mathcal {S}_d$ does not coincide with $\mathbb {C}^{d-1}$ for $d>2$.

In order to study the topology of the locus $\mathcal {S}_d$, Blanchard, Devaney and Keen (Blanchard et al. 1991) introduced the following homomorphism.³ Fix $p_{*}\in \mathcal {S}_d$, $p_{*}(z)=z^d+a_0$ with $|a_0|$ sufficiently large and choose a loop $\gamma : [0, 1]\rightarrow \mathcal {S}_d$ with $\gamma (0)=\gamma (1)=p_{*}$. Since $J_{\gamma (t)}$ is a Cantor set and $p=\gamma (t)$ is expanding on $J_p=J_{\gamma (t)}$ for all $t\in [0, 1]$, every point in $J_{\gamma (0)}=J_{p_{*}}$ uniquely continues to a point in $J_{\gamma (1)}=J_{p_{*}}$. In particular, this defines a homeomorphism $\rho _d(\gamma ) : J_{p_{*}}\rightarrow J_{p_{*}}$. It is easy to see that $\rho _d(\gamma )$ commutes with the shift map $\sigma$ on $\Sigma _d^{\mathbb {N}_0}$. Therefore, we have a homomorphism:

$$\begin{aligned} \rho _d : \pi _1(\mathcal {S}_d, p_{*})\longrightarrow \mathrm {Aut}(\Sigma _d^{\mathbb {N}_0}, \sigma ) \end{aligned}$$

satisfying $\rho _d(\gamma _1\cdot \gamma _2)=\rho _d(\gamma _2)\rho _d(\gamma _1)$. We call $\rho _d$ the monodromy representation of $\pi _1(\mathcal {S}_d, p_{*})$.

Theorem 1.22

(Blanchard et al. 1991) The monodromy representation $\rho _d$ is surjective for any $d\ge 2$.

The proof relies on the quasiconformal surgery of polynomials and the fact that an efficient system of generators for $\mathrm {Aut}(\Sigma _d^{\mathbb {N}_0}, \sigma )$ is known (Ashley 1990) (compare with the group of shift automorphisms $\mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma )$ on two-sided symbol sequences in Sect. 7.1).

(ix) Dynamics-parameter correspondence. We consider the quadratic family $p_c(z)=z^2+c$ ($c\in \mathbb {C}$) and discuss its dynamics-parameter correspondence discovered in Douady and Hubbard 1985 (see also Milnor 2000). Below we write $J_c\equiv J_{p_c}$, $K_c\equiv K_{p_c}$, $\varphi _c\equiv \varphi _{p_c}$, $R_c(\theta )\equiv R_{p_c}(\theta )$ and $G_c(z)\equiv G_{p_c}(z)$. Let $U_c\equiv \{z\in \mathbb {C} : G_c(z)>G_c(0)\}$. We then see that $U_c\cup \{\infty \}$ is a simply connected domain in the Riemann sphere for any $c\in \mathbb {C}$. Therefore, one can extend $\varphi _c$ to a holomorphic function:

$$\begin{aligned} \varphi _c : U_c\longrightarrow \mathbb {C}{\setminus } \overline{\Delta }. \end{aligned}$$

The Mandelbrot set $\mathcal {M}$ is defined as

$$\begin{aligned} \mathcal {M}\equiv \mathcal {M}_2=\big \{ c\in \mathbb {C} : J_c \text{ is } \text{ connected }\big \}=\big \{c\in \mathbb {C} : 0\in K_c \big \}. \end{aligned}$$

This leads to the following dichotomy.

• When $c\in \mathcal {M}$ (i.e. $J_c$ is connected), we have $G_c(0)=0$ by Corollary 1.2 and hence $U_c=\mathbb {C}{\setminus } K_c$. Therefore, $\varphi _c$ defines a conformal isomorphism $\varphi _c : \mathbb {C}{\setminus } K_c\rightarrow \mathbb {C}{\setminus } \overline{\Delta }$.

• When $c\notin \mathcal {M}$ (i.e. $J_c$ is disconnected), we have $G_c(0)>0$ by Corollary 1.2. This yields $G_c(c)=G_c(p_c(0))=2G_c(0)>G_c(0)$ and hence $c\in U_c$. In particular, $\varphi _c(c)$ is well-defined.

Theorem 1.23

(Douady and Hubbard 1985) The map:

$$\begin{aligned} \Phi : \mathbb {C}{\setminus } \mathcal {M}\ni c\longmapsto \varphi _c(c)\in \mathbb {C}{\setminus } \overline{\Delta } \end{aligned}$$

gives a conformal isomorphism. In particular, the Mandelbrot set $\mathcal {M}$ is connected.

People often call $\Phi (c)\equiv \varphi _c(c)$ the “magic formula” for the quadratic family $p_c$. Thanks to this theorem, we can define the external rays for $\mathcal {M}$.

Definition 1.24

We call $R_{\mathcal {M}}(\theta )\equiv \{\Phi ^{-1}(re^{2\pi i\theta }) : r>1\}$ the external ray of angle $\theta \in \mathbb {T}$ for the Mandelbrot set $\mathcal {M}$.

We say that an external ray $R_{\mathcal {M}}(\theta )$ lands on a point $c\in \partial \mathcal {M}$ if the limit $\lim _{r\downarrow 1}\Phi ^{-1}(re^{2\pi i\theta })$ exists and equals to c.

A connected component of $\{c\in \mathcal {M} : p_c \text{ is } \text{ expanding } \text{ on } J_c\}$ is called a hyperbolic component of $\mathcal {M}$. An example of a hyperbolic component is the Main Cardioid denoted by $\heartsuit$ consisting of the parameters $c\in \mathbb {C}$ so that $p_c$ has an attractive fixed point.

Let H be a hyperbolic component of $\mathcal {M}$ and let $c\in H$. By Theorem 1.6 we see that the orbit of 0 converges to a unique attractive cycle A of certain period $k(H)\ge 1$. Thanks to the chain rule, the multiplier $(p_c^{k(H)})'(a)$ of the cycle A is independent of the choice of $a\in A$.

Theorem 1.25

(Douady and Hubbard 1985) Let H be a hyperbolic component of $\mathcal {M}$. Then,

$$\begin{aligned} \Lambda _{H} : H \ni c\longmapsto (p_c^{k(H)})'(a)\in \Delta \end{aligned}$$

gives a conformal isomorphism.

We call $c(H)\equiv \Lambda _{H}^{-1}(0)\in H$ the center of H and $r_{\mathcal {M}}(H)\equiv \lim _{\lambda \uparrow 1}\Lambda _{H}^{-1}(\lambda )\in \overline{H}$ the root of H in the parameter space. Set

$$\begin{aligned} \Theta _{\mathcal {M}}(H)\equiv \big \{\theta \in \mathbb {T} : R_{\mathcal {M}}(\theta ) \text{ lands } \text{ on } r_{\mathcal {M}}(H)\big \}. \end{aligned}$$

Theorem 1.26

(Douady and Hubbard 1985) For any hyperbolic component H different from the Main Cardioid $\heartsuit$, there are exactly two angles $0<\theta ^-_{H}<\theta ^+_{H}<1$ so that $\Theta _{\mathcal {M}}(H)=\{\theta ^-_{H}, \theta ^+_{H}\}$.

For convenience we set $\theta ^-_{\heartsuit }\equiv 0$ and $\theta ^+_{\heartsuit }\equiv 1$ and therefore $\Theta _{\mathcal {M}}(\heartsuit )=\{0\}\subset \mathbb {T}$.

Now let us describe a surprising dynamics-parameter correspondence. Let H be a hyperbolic component of $\mathcal {M}$. Then, 0 is a superattractive periodic point of period k(H) for $p_{c(H)}$. Let $F_{c(H)}$ be the Fatou component of $p_{c(H)}$ containing the critical value c(H) of $p_{c(H)}$. By Böttcher’s theorem, there exists a unique conformal isomorphism:

$$\begin{aligned} \chi _{c(H)} : F_{c(H)}\longrightarrow \Delta \end{aligned}$$

which conjugates $p_{c(H)}^{k(H)} : F_{c(H)}\rightarrow F_{c(H)}$ to $\Delta \ni z\mapsto z^2\in \Delta$ and $\chi '_{c(H)}(c(H))=1$. We call $r(F_{c(H)})\equiv \lim _{z\uparrow 1}\chi _{c(H)}^{-1}(z)\in \overline{F_{c(H)}}$ the root of $F_{c(H)}$ in the dynamical space. Set

$$\begin{aligned} \Theta (F_{c(H)})\equiv \big \{\theta \in \mathbb {T} : R_{c(H)}(\theta ) \text{ lands } \text{ on } r(F_{c(H)})\big \}. \end{aligned}$$

The next claim builds a “bridge” between the dynamical space and the parameter space.

Theorem 1.27

(Douady and Hubbard 1985) For any hyperbolic component H of $\mathcal {M}$, we have $\Theta (F_{c(H)}) \supset \Theta _{\mathcal {M}}(H)$.

Here is a list of examples:

• When $H=\heartsuit$ is the Main Cardioid, we have $\Theta (F_{c(\heartsuit )})=\Theta _{\mathcal {M}}(\heartsuit )=\{0\}$.

• When H is the Basilica component, we have $\Theta (F_{c(H)})=\Theta _{\mathcal {M}}(H)=\{1/3, 2/3\}$.

• When H is the Airplane component, we have $\Theta (F_{c(H)})=\Theta _{\mathcal {M}}(H)=\{3/7, 4/7\}$.

• When H is the Rabbit component, we have $\{1/7, 2/7, 4/7\}=\Theta (F_{c(H)})\supset \Theta _{\mathcal {M}}(H)= \{1/7, 2/7\}$.

(x) Real quadratic family. In this item (x) we discuss an application of complex methods to real dynamics. A quadratic map $p_c$ is said to be real if $c\in \mathbb {R}$. A real map $p_c$ is called a hyperbolic horseshoe on $\mathbb {R}$ if the restriction of $p_c|_{\mathbb {R}} : \mathbb {R}\rightarrow \mathbb {R}$ to its non-wandering set is expanding and topologically conjugate to the shift map $\sigma : \Sigma _2^{\mathbb {N}_0}\rightarrow \Sigma _2^{\mathbb {N}_0}$. We also know that $0\le h_{\mathrm {top}}(p_c|_{\mathbb {R}})\le \log 2$ holds for all $c\in \mathbb {R}$. Therefore, we say that $p_c$ attains the maximal entropy on $\mathbb {R}$ if $h_{\mathrm {top}}(p_c|_{\mathbb {R}})=\log 2$.

Theorem 1.28

A real quadratic map $p_c$ is a hyperbolic horseshoe on $\mathbb {R}$ iff $c<-2$, and $p_c$ attains the maximal entropy on $\mathbb {R}$ iff $c\le -2$.

In particular, the boundary of the hyperbolic horseshoe locus and the maximal entropy locus for $p_c$ coincide and equal to the one-point set $\{-2\}$. The proof of the above theorem is supplied by using Poincaré metrics and the symmetry of $p_c$ with respect to the complex conjugation.

2 Preliminaries on Polynomial Diffeomorphisms of $\mathbb {C}^2$

In this section we recall some preliminaries on polynomial diffeomorphisms of $\mathbb {C}^2$ and propose ten problems related to the ten items presented in the previous section.

2.1 Classification

A polynomial map $f : \mathbb {C}^2\rightarrow \mathbb {C}^2$ is called a polynomial diffeomorphim of $\mathbb {C}^2$ if it has a polynomial inverse. Examples of polynomial diffeomorphims of $\mathbb {C}^2$ are an affine map:

$$\begin{aligned} \alpha : (x, y)\longmapsto (a_1 x+b_1 y+c_1, a_2 x+b_2 y+c_2) \end{aligned}$$

where $a_1b_2-a_2b_1\ne 0$, an elementary map:

$$\begin{aligned} \beta : (x, y)\longmapsto (ax+c, p(x)+by) \end{aligned}$$

where p(x) is a polynomial of degree $d\ge 2$ and $ab\ne 0$, and a generalized Hénon map:

$$\begin{aligned} f_{p, b} : (x, y)\longmapsto (p(x)-by, x) \end{aligned}$$

where p(x) is a polynomial of degree $d\ge 2$ and $b\in \mathbb {C}^{\times }\equiv \mathbb {C}{\setminus } \{0\}$.

Let $\mathrm {Poly}(\mathbb {C}^2)$ be the space of polynomial diffeomorphisms of $\mathbb {C}^2$. This forms a group by the composition of two maps and the conjugacy classes can be classified into three types.

Theorem 2.1

(Friedland and Milnor 1989) Any $f\in \mathrm {Poly}(\mathbb {C}^2)$ is conjugate in the group $\mathrm {Poly}(\mathbb {C}^2)$ to either

• (1) an affine map,

• (2) an elementary map, or

• (3) a composition of finitely many generalized Hénon maps.

The proof of Theorem 2.1 is based on a classical result of Jung (1942) which claims that the group $\mathrm {Poly}(\mathbb {C}^2)$ is generated by the affine maps and the elementary maps. One may wonder if an analogous result holds for the group of polynomial diffeomorphisms of $\mathbb {C}^3$. In his 1972 paper (Nagata 1972), Masayoshi Nagata⁴ proposed the map:

$$\begin{aligned} (x, y, z)\longmapsto (x+(x^2-yz)z, y+2(x^2-yz)x+(x^2-yz)^2z, z) \end{aligned}$$

as a possible counterexample to this analogy. More than 30 years later, Shestakov and Umirbaev (2004) finally showed Nagata’s conjecture in the affirmative, i.e. the Nagata map is not contained in the subgroup of $\mathrm {Poly}(\mathbb {C}^3)$ generated by affine maps and the elementary maps.

It is easy to see that the dynamics of the cases (a) and (b) in Theorem 2.1 is simple, so the only dynamically interesting case is (c). Therefore, we will hereafter treat a map of the form:

$$\begin{aligned} f=f_{p_1, b_1}\circ \cdots \circ f_{p_k, b_k}. \end{aligned}$$

Note that for a map of this form, the Jacobian determinant is given by $\det (Df)=b_1\cdots b_k$. We define $d\equiv d_1\cdots d_k$ and call it the degree of f, where $d_i\equiv \deg p_i$. The next result indicates that the maps in this class exhibit rich dynamics.

Theorem 2.2

(Friedland and Milnor 1989; Smillie 1990) We have $h_{\mathrm {top}}(f)=\log d$.

Let us define the forward/backward filled-Julia sets of f as

$$\begin{aligned} K^{\pm }\equiv \big \{(x, y)\in \mathbb {C}^2 : \{f^{\pm n}(x, y)\}_{n\ge 0} \text{ is } \text{ bounded } \big \}, \end{aligned}$$

the forward/backward Julia sets of f as $J^{\pm }\equiv \partial K^{\pm }$. We also define $K_f\equiv K^+\cap K^-$. We put $J=J_f\equiv J^+\cap J^-$ and call it the Julia set ⁵ of f.

As a comparison with the quadratic family $p_c$ we consider the complex Hénon family:

$$\begin{aligned} f_{c, b} : (x, y)\longmapsto (x^2+c-by, x) \end{aligned}$$

defined on $\mathbb {C}^2$, where $(c, b)\in \mathbb {C}\times \mathbb {C}^{\times }$ is a parameter. Let us call $f_{c, b}$ real if $(c, b)\in \mathbb {R}\times \mathbb {R}^{\times }$. When $f_{c, b}$ is real, the dynamical system $f_{c, b} : \mathbb {R}^2\rightarrow \mathbb {R}^2$ is well-defined.

2.2 Ten Problems

Based on the ten items discussed in Sect. 1, we propose the following ten problems for polynomial diffeomorphisms f of $\mathbb {C}^2$ or the Hénon family.

Problems:

• (i) Define “dynamical critical points” of f. Related it to the connectivity of the Julia set.

• (ii) When the Julia set is connected, define the notion of external rays.

• (iii) Establish a criterion for hyperbolicity of f on the Julia set and construct examples.

• (iv) When the Julia set is connected and hyperbolic, describe it as a quotient space of a “simple” space like a circle.

• (v) Define the notion of a Hubbard tree for f. Prove that it reconstructs the Julia set.

• (vi) Construct a (tight) automaton for f. Prove that it reconstructs the Julia set.

• (vii) Define the notion of an iterated monodromy group for f. Prove that its limit space is homeomorphic to the Julia set.

• (viii) Study the monodromy representation for the complex Hénon family. Is it surjective?

• (ix) Establish a dynamics-parameter correspondence for the complex Hénon family.

• (x) Characterize the hyperbolic horseshoe locus and the maximal entropy locus for the real Hénon family.

In the rest of this article, we will discuss the above problems for polynomial diffeomorphisms of $\mathbb {C}^2$ or for the Hénon family. Section 3 is devoted to Problems (i), (ii) and (iv) where the results are obtained in Bedford and Smillie (1998a, b, 1999). Section 4 is the “Intermezzo” of this article and noting to do with the problem list above, where we discuss an application of the convergence theorem of currents (Bedford and Smillie 1991a, b, 1992) to curious objects in general topology called the Lakes of Wada. Section 5 discusses Problem (iii) and we present a construction of a hyperbolic Hénon map with intrinsically two-dimensional dynamics in Ishii (2008). The following Section 6 is dedicated to Problems (v)–(vii) where we present some results in Ishii (2009, 2014). Problems (viii) and (ix) are discussed in Sect. 7 and two conjectures from Lipa (2009) are presented. Finally in Sect. 8 we consider Problem (x) and present some related results in Bedford and Smillie (2004, 2006), Arai and Ishii (2015), Arai et al. (2017).

3 Connectivity of the Julia Sets and their External Rays

This section is devoted to Problems (i)–(iii). Recall that $J_f=J_{f^{-1}}$ holds. Therefore, so far as we discuss the connectivity of $J_f$, we may assume $|\det (Df)| \le 1$.

3.1 Connectivity

We first introduce the following notion.

Definition 3.1

We say that the Julia set $J_f$ is unstably connected with respect to a saddle periodic point q if $W^u(q)\cap J_f$ has no compact components.

The following fundamental theorem states that the connectivity of $J_f$ can be detected through the complex one-dimensional slice of $J_f$ by some/any unstable manifold.

Theorem 3.2

(Bedford and Smillie 1998b) Let $|\det (Df)| \le 1$. Then, the following are equivalent:

• (1) $J_f$ is connected,

• (2) $J_f$ is unstably connected with respect to some saddle periodic point q,

• (3) $J_f$ is unstably connected with respect to any saddle periodic point q.

Indeed, Bedford and Smillie showed (Theorem 0.1 in Bedford and Smillie 1998b) that (2) and (3) above are equivalent without $|\det (Df)| \le 1$, and called a map f satisfying the conditions unstably connected.

Our next task is to restate the conditions (2) and (3) in the theorem above so that they can be verified by computer experiments. To do this, let us introduce the Green functions of f as

$$\begin{aligned} G^{\pm }(x, y)\equiv \lim _{n\rightarrow +\infty }\frac{1}{d^n}\log ^+\Vert f^{\pm n}(x, y)\Vert . \end{aligned}$$

One can see that $G^{\pm }(x, y)$ are continuous and plurisubharmonic and satisfy $G^{\pm }(f(x, y))=d^{\pm 1}\cdot G^{\pm }(x, y)$ on $\mathbb {C}^2$, pluriharmonic on $\mathbb {C}^2{\setminus } K^{\pm }$ and $G^{\pm }(x, y)>0$ iff $(x, y)\in \mathbb {C}^2{\setminus } K^{\pm }$. Therefore, $\mu ^{\pm }\equiv \frac{1}{2\pi }dd^cG^{\pm }$ define positive (1, 1)-currents on $\mathbb {C}^2$.

Define an analogy of the Böttcher coordinate:

$$\begin{aligned} \varphi ^+(x, y)\equiv \lim _{n\rightarrow +\infty }(\pi _x\circ f^n(x, y))^{\frac{1}{d^n}} \end{aligned}$$

(by choosing an appropriate $d^n$-th root) for $(x, y)\in V_R^+\equiv \{(x, y)\in \mathbb {C}^2 : |x|>R, \, |x|>|y|\}$, where $\pi _x$ is the projection to the x-axis and $R>0$ large. Note that we have $\varphi ^+(x, y)/x\rightarrow 1$ as $|x|\rightarrow \infty$ for every fixed y and $G^+(x, y)=\log |\varphi ^+(x, y)|$ for $(x, y)\in V_R^+$.

It was observed in Bedford and Smillie (1998b) that, when we try to extend $\varphi ^+$ along $J^-{\setminus } K^+$, an obstruction is the critical points of $G^+$ on $W^u(q){\setminus } K^+$. This leads to define the dynamical critical set as

$$\begin{aligned} \mathcal {C}^u\equiv \bigcup _{q\in \mathcal {R}}\mathrm {Crit}(G^+; q), \end{aligned}$$

where $\mathcal {R}$ denotes the set of Pesin regular points in $J_f$ (e.g. the saddle periodic points) and

$$\begin{aligned} \mathrm {Crit}(G^+; q)\equiv \big \{(x, y)\in W^u(q){\setminus } K^+ : (x, y) \text{ is } \text{ a } \text{ critical } \text{ point } \text{ of } G^+|_{W^u(q){\setminus } K^+} \big \} \end{aligned}$$

for $q\in \mathcal {R}$ (see Bedford and Smillie 1998a). These critical points represent tangencies between the lamination of $J^-$ by unstable manifolds and the foliation on $\mathbb {C}^2{\setminus } K^+$ defined by the holomorphic 1-form $\partial G^+$. By the laminar structure of $\mu ^-$, it induces a measure $\mu ^-_c$ on the dynamical critical set $\mathcal {C}^u$. This measure yields a formula for the Lyapunov exponent of f (Bedford and Smillie 1998a):

$$\begin{aligned} \Lambda _{\mu }(f)=\log d+\int _{\{1\le G^+<d\}}G^+d\mu ^-_c \end{aligned}$$

with respect to the unique maximal entropy measure $\mu \equiv \mu ^+\wedge \mu ^-$ (see Bedford et al. 1993a for more details), where $\{1\le G^+<d\}$ is a fundamental domain for $\mathcal {C}^u$. This formula generalizes the corresponding one-dimensional formula given in Przytycki (1985) and was a key step in the proof of Theorem 3.2.

The following claim has been obtained as a combination of an argument by Dujardin and ones in Bedford and Smillie (1998b). For the proof we refer to Ishii (2011).

Theorem 3.3

(Bedford and Smillie 1998b, Dujardin) Let $|\det (Df)| \le 1$ and let q be any saddle periodic point of f. Then, the following holds.

• (1) $J_f$ is connected iff $K_f$ is connected.

• (2) $J_f$ is unstably connected with respect to q iff $\mathrm {Crit}(G^+; q)=\emptyset$.

In particular, Theorems 3.2 and 3.3 yield

Corollary 3.4

Let $|\det (Df)| \le 1$ and let q be any saddle periodic point of f. Then, the Julia set $J_f$ is connected iff $\mathrm {Crit}(G^+; q)=\emptyset$.

This justifies the algorithm of the program SaddleDrop (SaddleDrop 2000) to draw the connectedness locus of the complex Hénon family. SaddleDrop was written around 2000 by Karl Papadantonakis, then an undergraduate student at Cornell. The procedure to use SaddleDrop is as follows.

• Step 1: Choose $(c_0, b_0)\in \mathbb {C}\times \mathbb {C}^{\times }$ with $|b_0|\le 1$.

• Step 2: Compute $W^u(q)$ of a saddle fixed point q for $f_{c_0, b_0}$.

• Step 3: Draw the set $K^+\cap W^u(q)$ as well as some equi-potential curves of $G^+|_{W^u(q)}$ for $f_{c_0, b_0}$ in the uniformized coordinate $\mathbb {C}\cong W^u(q)$.

• Step 4: By looking at equi-potential curves of $G^+|_{W^u(q)}$ in $\mathbb {C}\cong W^u(q)$, we try to find an element in $\mathrm {Crit}(G^+; q)$.

• Step 5: If we can find a critical point, click it; then SaddleDrop traces all parameters $(c, b_0)\in \mathbb {C}\times \mathbb {C}^{\times }$ to which a continuation of the chosen critical point survives. Repeat this procedure for as many points in $\mathrm {Crit}(G^+; q)$ as you can find.

• Step 6: Choose a new $c_0\in \mathbb {C}$ (by keeping $b_0\in \mathbb {C}^{\times }$) which was not traced from any of the previous choices and return to Step 2.

According to Corollary 3.4, the algorithm above yields the following claims.

• (i) If $(c, b_0)$ can be traced from some $(c_0, b_0)$ through a critical point, then we are sure (up to numerical error) that the Julia set of $f_{c, b_0}$ is disconnected.

• (ii) If $(c, b_0)$ can not be traced from any $(c_0, b_0)$ through any critical points we found, then the Julia set of $f_{c, b_0}$ is “presumably” connected.

The claim (ii) is valid at best “presumably” because we do not know if the intersection of the complement of the connectedness locus with the slice $\{(c, b)\in \mathbb {C}\times \mathbb {C}^{\times } : b=b_0\}$ is connected and because it is not theoretically possible to do Step 5 in the algorithm for all points in $\mathrm {Crit}(G^+; q)$ which could be an infinite set. Modulo these issues, it seems that SaddleDrop may give a good approximation of the connectedness locus for the complex Hénon family. We refer to Koch (2010) for several pictures obtained by SaddleDrop as well as some other issues related to the algorithm.

3.2 External Rays

In this section we define the notion of external rays and discuss a topological model for a connected and hyperbolic Julia set. We say that a polynomial diffeomorphism f of $\mathbb {C}^2$ is hyperbolic if its Julia set $J_f$ is a hyperbolic set for f.

Let $p_d : \mathbb {C}{\setminus }\overline{\Delta }\rightarrow \mathbb {C}{\setminus }\overline{\Delta }$ be the monomial map $p_d(z)\equiv z^d$ of degree $d\ge 2$. The projective limit of $p_d : \mathbb {C}{\setminus }\overline{\Delta }\rightarrow \mathbb {C}{\setminus }\overline{\Delta }$ denoted as $\mathbb {S}_d^{\mathbb {C}}$ together with the shift map on it $\hat{p}_d : \mathbb {S}_d^{\mathbb {C}}\rightarrow \mathbb {S}_d^{\mathbb {C}}$ is called the complex solenoid of degree d (denoted as $\Sigma _+$ in Bedford and Smillie 1999). Similarly, let $\delta _d : \mathbb {T}\rightarrow \mathbb {T}$ be the map given by $\delta _d(\theta )\equiv d\cdot \theta$. The projective limit of $\delta _d : \mathbb {T}\rightarrow \mathbb {T}$ denoted as $\mathbb {S}_d^{\mathbb {R}}$ together with the shift map on it $\hat{\delta }_d : \mathbb {S}_d^{\mathbb {R}}\rightarrow \mathbb {S}_d^{\mathbb {R}}$ is called the real solenoid of degree d (denoted as $\Sigma _0$ in Bedford and Smillie 1999). In the context of complex Hénon maps, real solenoids had appeared earlier, although in quite a different form, in Hubbard (1986), Hubbard and Oberste-Vorth (1994) as “the dynamics at infinity”.

The next theorem states that, when $J_f$ is connected and f is hyperbolic, one can define the notion of external rays in $J^-{\setminus } K^+$ which are parameterized by the “space of angles” $\mathbb {S}_d^{\mathbb {R}}$.

Theorem 3.5

(Bedford and Smillie 1999) Let $|\det (Df)| \le 1$. If $J_f$ is connected and f is hyperbolic, there exists a homeomorphism:

$$\begin{aligned} \Psi \, : \, \mathbb {S}_d^{\mathbb {C}} \longrightarrow J^-{\setminus } K^+ \end{aligned}$$

which conjugates the shift map $\hat{p}_d : \mathbb {S}_d^{\mathbb {C}}\rightarrow \mathbb {S}_d^{\mathbb {C}}$ to $f : J^-{\setminus } K^+ \rightarrow J^-{\setminus } K^+$.

Indeed, it is shown in Bedford and Smillie (1998b) that, if $|\det (Df)| \le 1$ and $J_f$ is connected, the holomorphic function $\varphi ^+ : V_R^+\rightarrow \mathbb {C}{\setminus } \{|z|\le R\}$ extends to $\varphi ^+ : J^-{\setminus } K^+\rightarrow \mathbb {C}{\setminus } \overline{\Delta }$. Hence one can define

$$\begin{aligned} \Phi \, : \, J^-{\setminus } K^+\ni (x, y)\longmapsto (\varphi ^+\circ f^n(x, y))_{n\in \mathbb {Z}}\in \mathbb {S}_d^{\mathbb {C}}. \end{aligned}$$

When moreover f is hyperbolic, $\Phi : J^-{\setminus } K^+\rightarrow \mathbb {S}_d^{\mathbb {C}}$ is a finite covering (Bedford and Smillie 1999). By modifying the “local inverse map” of $\Phi$ appropriately, we obtain the homeomorphism $\Psi : \mathbb {S}_d^{\mathbb {C}} \rightarrow J^-{\setminus } K^+$ in Theorem 3.5 (see Section 4 in Bedford and Smillie 1999). It is still an open question if $\Phi$ itself is a homeomorphism (see a remark just after Corollary 4.2 of Bedford and Smillie 1999). If it is the case, we have $\Psi =\Phi ^{-1}$.

Thanks to the theorem above one can define (when f is hyperbolic) the notion of external rays in $J^-{\setminus } K^+$ as the push-forward of the rays in $\mathbb {S}_d^{\mathbb {C}}$ by $\Psi$. Hyperbolicity of f also implies that every external ray has a well-defined landing point in $J_f$. In particular, we have

Corollary 3.6

(Bedford and Smillie 1999) Let $|\det (Df)| \le 1$. If $J_f$ is connected and f is hyperbolic, $\Psi$ extends to a surjective semiconjugacy $\Psi : \mathbb {S}_d^{\mathbb {R}} \rightarrow J_f$ from $\hat{\delta }_d : \mathbb {S}_d^{\mathbb {R}}\rightarrow \mathbb {S}_d^{\mathbb {R}}$ to $f : J_f\rightarrow J_f$.

In particular, $f : J_f\rightarrow J_f$ is conjugate to the factor $\hat{\delta }_d/_{\sim _{\mathrm {BS}}} : \mathbb {S}_d^{\mathbb {R}}/_{\sim _{\mathrm {BS}}}\rightarrow \mathbb {S}_d^{\mathbb {R}}/_{\sim _{\mathrm {BS}}}$, where we define $\underline{\theta }\sim _{\mathrm {BS}} \underline{\theta }'$ iff $\Psi (\underline{\theta })=\Psi (\underline{\theta }')$ for $\underline{\theta }, \underline{\theta }'\in \mathbb {S}_d^{\mathbb {R}}$. The nature of $\sim _{\mathrm {BS}}$ is, however, still mysterious (cf. the thesis of Oliva 1998) and we will discuss this issue in Sect. 6.

4 Fatou–Bieberbach Domains and Lakes of Wada in $\mathbb {C}^2$

In this survey article we focus on combinatorial and topological aspects of the dynamics. An important topic we miss here is potential theoretic and ergodic approach which has been extensively studied in a series of papers (Bedford and Smillie 1991a, b, 1992; Bedford et al. 1993a, b) and also in Fornæss and Sibony (1992).

4.1 Lakes of Wada

First we remark that as a consequence of the convergence theorem of currents, Bedford and Smillie obtained the following curious result.

Theorem 4.1

(Bedford and Smillie 1991b) For any attractive basin B for f, we have $\partial B=J^+$.

This result reminds us the so-called Lakes of Wada; mutually disjoint three domains in the plane which possess common boundary. Such a curious example was first constructed in an article of Yoneyama (1917). Here we quote the following nice explanation from Hubbard and Oberste-Vorth (1995):

Consider a circular island, inhabited, to the sorrow of the others, by three philanthropists. One has a lake of water, another of milk and a third of wine. The first, in a fit of generosity, decides to built a network of canals bringing water within 100 m of every spot of the island. It is clearly possible to do this keeping the union of the original water lake and the water canals connected and simply connected, with closures disjoint from the other lakes.

Next the second, perhaps worried about child nutrition, decides to bring milk to within 10 m of every spot on the island, and builds a system of canals to that effect. She also keeps her milk locus connected and simply connected.

Not to be outdone, the purveyor of wine now decides to bring wine to within 1 m of every spot on the island. He finds his canal building rather more of an effort than the previous two, but being properly fortified, he carries it out.

In turn, each of the three philanthropists brings his or her product closer to the poor inhabitants. It should be clear that the construction can be continued, and that in the limit the construction achieves the desired result: each of the lakes, being an increasing union of connected and simply connected open sets, is a connected, simply connected set, and each point of the boundary of one is in the boundary of the other two.

In the same paper Yoneyama writes that the construction “was informed to me by Mr. Wada.” (see the footnote in page 60 of Yoneyama 1917). This is why such domains are now called Lakes of Wada. But, who is this Mr. Wada? Here is his picture (Fig. 8).⁶

He is Takeo Wada, a Japanese mathematician working on analysis and general topology. He was one of the first students in the mathematics department of Kyoto Imperial University (now Kyoto University) and he later became a full professor there.⁷

Fig. 8

Takeo Wada (1882–1944)

What is surprising about Theorem 4.1 is that a single map f possessing at least two attractive periodic points generates such domains in $\mathbb {C}^2$ (notice that the boundary of $\mathbb {C}^2{\setminus } K^+$ is also $J^+$). In Hubbard and Oberste-Vorth (1995) obtained a sufficient condition for a real Hénon map to have such curious domains in $\mathbb {R}^2$ as its attractive basins.

4.2 FB Domains

A Fatou–Bieberbach domain in $\mathbb {C}^2$ is a proper subdomain in $\mathbb {C}^2$ which is biholomorphically equivalent to $\mathbb {C}^2$. Since any attractive basin of f is contained in $K^+$ and since $\mathbb {C}^2{\setminus } K^+$ is non-empty, it is always a Fatou–Bieberbach domain.

Another question related to Theorem 4.1 is the existence/non-existence of a Fatou–Bieberbach domains with smooth boundaries. There exists a Fatou–Bieberbach domain with $C^{\infty }$-smooth boundary by using a non-autonomous iterations (Stensønes 1997). On the other hand, as a consequence of Theorem 4.1 we have

Corollary 4.2

(Bedford and Smillie 1991b) If f has at least two attractive basins, their boundaries cannot even be a topological manifold at any point.

Therefore, the only remaining case where the boundary of an attractive basin could be smooth is when f has only one attractive basin. Very recently the following result has appeared.

Theorem 4.3

(Bedford and Kim 2015) For any f, its forward Julia set $J^+$ can not be smooth of class $C^1$ as a manifold with boundary.

This result gives a complete answer to the question mentioned above.

5 Construction of Hyperbolic Complex Hénon Maps

In this section we discuss Problem (iii). In Sect. 5.1 we establish a criterion for hyperbolicity of a polynomial diffeomorphism of $\mathbb {C}^2$ and in Sect. 5.2 we construct a first example of a hyperbolic Hénon map whose dynamics is intrinsically complex two-dimensional.

5.1 Hyperbolicity

Let $p : \mathbb {C}\rightarrow \mathbb {C}$ be a polynomial of $\deg p\ge 2$ and let $J_p$ is its Julia set. Following (Hubbard and Oberste-Vorth 1995) we denote by $\hat{J}_p\equiv \varprojlim (p, J_p)$ the projective limit of $p : J_p\rightarrow J_p$ and by $\hat{p} : \hat{J}_p\rightarrow \hat{J}_p$ the shift map on it.

Definition 5.1

A polynomial diffeomorphism f of $\mathbb {C}^2$ is called planar ⁸ if there exists an expanding polynomial p so that $f : J_f\rightarrow J_f$ is topologically conjugate to $\hat{p} : \hat{J}_p\rightarrow \hat{J}_p$.

As an example of hyperbolic polynomial diffeomorphisms of $\mathbb {C}^2$, it is known that a small perturbation of an expanding polynomial is hyperbolic. More precisely,

Theorem 5.2

Let p be expanding. Then, there exists $b_{*}>0$ so that for any $0<|b|<b_{*}$ the generalized Hénon map $f=f_{p, b}$ is hyperbolic on its Julia set (Fornæss and Sibony 1992; Hubbard and Oberste-Vorth 1995). Moreover, $f : J_f\rightarrow J_f$ is topologically conjugate to $\hat{p} : \hat{J}_p\rightarrow \hat{J}_p$, i.e. such $f_{p, b}$ is planar (Hubbard and Oberste-Vorth 1995).

Next we introduce a criterion for hyperbolicity of polynomial diffeomorphisms f of $\mathbb {C}^2$. Let $A_x$ and $A_y$ be bounded domains in $\mathbb {C}$ and let $\mathcal {A}=A_x\times A_y$. We then have projections $\pi _x : \mathcal {A}\rightarrow A_x$ and $\pi _y : \mathcal {A}\rightarrow A_y$. The following condition has been first introduced in Hubbard and Oberste-Vorth (1995) when $\mathcal {A}$ is a polydisk (see Ishii 2008; Ishii and Smillie 2010 for more general case).

Definition 5.3

We call $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ a crossed mapping of degree d if

$$\begin{aligned} \rho _f\equiv (\pi _x \circ f, \pi _y \circ \iota _{\mathcal {A}}) \, : \, \mathcal {A}\cap f^{-1}(\mathcal {A})\longrightarrow \mathcal {A} \end{aligned}$$

is proper of degree d, where $\iota _{\mathcal {A}} : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ is the inclusion.

Let $\mathcal {F}_h=\{A_x(y)\}_{y\in A_y}$ be the horizontal foliation of $\mathcal {A}\cap f^{-1}(\mathcal {A})$ with leaves $A_x(y)=(A_x\times \{y\})\cap (\mathcal {A}\cap f^{-1}(\mathcal {A}))$, and $\mathcal {F}_v=\{A_y(x)\}_{x\in A_x}$ be the vertical foliation of $\mathcal {A}\cap f^{-1}(\mathcal {A})$ with leaves $A_y(x)=(\{x\}\times A_y)\cap (\mathcal {A}\cap f^{-1}(\mathcal {A}))$. Another condition we employ is the following (Ishii 2008; Ishii and Smillie 2010).

Definition 5.4

We say that a crossed mapping $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ satisfies the no-tangency condition if $f(\mathcal {F}_h)$ and $\iota _{\mathcal {A}}(\mathcal {F}_v)$ have no tangencies in $\mathcal {A}$.

Let $|\cdot |_{A_x}$ and $|\cdot |_{A_y}$ be Poincaré metrics in $A_x$ and $A_y$ respectively. The horizontal Poincaré cone field $(\{C^h_q\}_{q\in \mathcal {A}}, \Vert \cdot \Vert _h)$ is

$$\begin{aligned} C^h_q\equiv \bigl \{v=(v_x, v_y) \in T_q \mathcal {A} : |v_x|_{A_x}\ge |v_y|_{A_y} \bigr \} \end{aligned}$$

with the metric $\Vert v\Vert _h\equiv |D\pi _x (v)|_{A_x}$. The vertical Poincaré cone field $(\{C^v_q\}_{q\in \mathcal {A}}, \Vert \cdot \Vert _v)$ is

$$\begin{aligned} C^v_q\equiv \bigl \{v=(v_x, v_y) \in T_q \mathcal {A} : |v_x|_{A_x}\le |v_y|_{A_y} \bigr \} \end{aligned}$$

with the metric $\Vert v\Vert _v\equiv |D\pi _y (v)|_{A_y}$. A product set $\mathcal {A}=A_x\times A_y$ equipped with the horizontal and the vertical Poincaré cone fields is called a Poincaré box.

A crossed mapping $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ is said to expand the horizontal Poincaré cone field if there exists $\lambda >1$ so that for any $q\in \mathcal {A}\cap f^{-1}(\mathcal {A})$, we have $D\iota _{\mathcal {A}}^{-1}(C^h_{\iota _{\mathcal {A}}(q)})\subset Df^{-1}(C^h_{f(q)})$ and $\lambda \Vert D\iota _{\mathcal {A}} (v)\Vert _h\le \Vert Df(v)\Vert _h$ for any $v\in T_q(\mathcal {A}\cap f^{-1}(\mathcal {A}))$ with $D\iota _{\mathcal {A}}(v)\in C^h_{\iota _{\mathcal {A}}(q)}$. Similarly, a crossed mapping is said to contract the vertical Poincaré cone field if there exists $\lambda >1$ so that for any $q\in \mathcal {A}\cap f^{-1}(\mathcal {A})$, we have $Df^{-1}(C^v_{f(q)})\subset D\iota _{\mathcal {A}}^{-1}(C^v_{\iota _{\mathcal {A}}(q)})$ and $\lambda \Vert Df(v)\Vert _v\le \Vert D\iota _{\mathcal {A}} (v)\Vert _v$ for any $v\in T_q(\mathcal {A}\cap f^{-1}(\mathcal {A}))$ with $Df(v)\in C^v_{f(q)}$. A crossed mapping is called a hyperbolic system if it expands the horizontal Poincaré cone field and contracts the vertical Poincaré cone field.

The following statements give hyperbolicity criterion for polynomial diffeomorphisms of $\mathbb {C}^2$ with a single Poincaré box.

Theorem 5.5

(Ishii 2008; Ishii and Smillie 2010) If a crossed mapping $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ satisfies the non-tangency condition, it is a hyperbolic system.

Corollary 5.6

(Ishii 2008; Ishii and Smillie 2010) If a crossed mapping $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ satisfies the non-tangency condition, f is uniformly hyperbolic on $\bigcap _{n\in \mathbb {Z}}f^n(\mathcal {A})$.

Note that there are more checkable criteria for a map to be a crossed mapping called the boundary compatibility condition (Definition 2.15 in Ishii 2008) and for the non-tangency condition called the off-criticality condition (Definition 2.16 in Ishii 2008) which can be verified by hand or by computer assistance. Together with the technique of homotopy shadowing developed in Ishii and Smillie (2010), one obtains a quantitative estimate for the constant $b_{*}$ in Theorem 5.2 and a new proof of the latter half of the statement in the theorem.

Corollary 5.7

For the complex Hénon family:

$$\begin{aligned} f=f_{c, b} : (x, y)\longmapsto (x^2+c-by, x), \end{aligned}$$

we have the following.

• (1) When $|c|>2$, we may take $b_{*}=\sqrt{|c|/2}-1$ in Theorem 5.2 (Oberste-Vorth 1987; Morosawa et al. 2000).

• (2) When $c=0$, we may take $b_{*}=(\sqrt{2}-1)/2$ in Theorem 5.2 (Ishii and Smillie 2010).

• (3) When $c=-1$, we may take $b_{*}=0.02$ in Theorem 5.2 (Ishii and Smillie 2010).

We remark that Theorem 5.5 as well as Corollary 5.6 holds for a map f from a Poincaré box to a different Poincaré box (see Corollary 2.17 in Ishii 2008). Moreover, one can extend them to the case where f is a system of maps from the disjoint union of finitely many Poincaré boxes to itself (see Corollary 2.18 in Ishii 2008).

Fig. 9

Poincaré boxes for the non-planar map in Theorem 5.8

5.2 Non-planarity

Theorem 5.2 tells that the dynamics of a Hénon map obtained as a small perturbation of an expanding polynomial is intrinsically complex one-dimensional. Therefore, a natural question arise; is there a hyperbolic polynomial diffeomorphism which is non-planar? The first non-planar example of a hyperbolic Hénon map was constructed by the author (Ishii 2008).

Theorem 5.8

(Ishii 2008) The cubic complex Hénon map:

$$\begin{aligned} f_{a, b} : (x, y)\longmapsto (-x^3+a-by, x) \end{aligned}$$

with $(a, b)=(-1.35, 0.2)$ is hyperbolic but non-planar.

The proof of Theorem 5.8 goes as follows. First we choose four Poincaré boxes $\{\mathcal {A}_i\}_{i=0}^3$ in $\mathbb {C}^2$ where $\mathcal {A}_i=A_{x, i}\times A_{y, i}$ (see Fig. 9). Note that $A_{x, 0}$ and $A_{x, 3}$ are annuli so that $\mathcal {A}_0$ and $\mathcal {A}_3$ have vertical holes which are shaded in Fig. 9. Also note that, $\mathcal {A}_1$ and $\mathcal {A}_2$ are drawn at the same place to simplify the figure, although they are actually disjoint. Set $\Sigma ^0\equiv \{0, 1, 2, 3\}$ and

$$\begin{aligned} \Sigma ^1\equiv \big \{(0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2)\big \}. \end{aligned}$$

We first show that the union $\bigcup _{i\in \Sigma ^0}\mathcal {A}_i$ covers the Julia set $J_f$ of the non-planar map $f=f_{a, b}$ in Theorem 5.8 with computer assistance. Next, the multivalued dynamical system:

$$\begin{aligned} \iota _{\mathcal {A}}, f \ : \ \mathcal {A}^1\equiv \bigcup _{(i, j)\in \Sigma ^1}\mathcal {A}_i\cap f^{-1}(\mathcal {A}_j) \ \longrightarrow \ \mathcal {A}^0\equiv \bigcup _{j\in \Sigma ^0}\mathcal {A}_j \end{aligned}$$

(1)

lifts to the disjoint union of boxes:

$$\begin{aligned} \tilde{\iota }_{\mathcal {A}}, \tilde{f} \ : \ \widetilde{\mathcal {A}}^1\equiv \bigsqcup _{(i, j)\in \Sigma ^1}\mathcal {A}_i\cap \tilde{f}^{-1}(\mathcal {A}_j) \ \longrightarrow \ \widetilde{\mathcal {A}}^0\equiv \bigsqcup _{j\in \Sigma ^0}\mathcal {A}_j. \end{aligned}$$

(2)

We show that $\iota _{\mathcal {A}}, f : \mathcal {A}_i\cap f^{-1}(\mathcal {A}_j)\rightarrow \mathcal {A}_j$ is a crossed mapping satisfying the non-tangency condition for all $(i, j)\in \Sigma ^1$ by verifying the checkable conditions mentioned above again by computer assistance. This implies that (2) is a hyperbolic system in an extended sense.

Note that since the boxes $\{\mathcal {A}_i\}_{i=0}^3$ have overlaps in $\mathbb {C}^2$, this does not immediately imply the hyperbolicity of the original system (1). To overcome this point, we define a new horizontal cone at a point in the overlap as

$$\begin{aligned} C_p^{\cap }\equiv \bigcap _{i\in I(p)}C^i_p \end{aligned}$$

for $p\in \mathcal {A}^0$, where $C^i_p$ is the horizontal Poincaré cone at $p\in \mathcal {A}_i$ with respect to the Poincaré box $\mathcal {A}_i$ and $I(p)\equiv \left\{ i\in \Sigma ^0 : p\in \mathcal {A}_i\right\}$. We also define a metric $\Vert \cdot \Vert _{\cap }$ in it by

$$\begin{aligned} \Vert v\Vert _{\cap }\equiv \min \big \{\Vert v\Vert _{\mathcal {A}_i} : i\in I(p) \big \} \end{aligned}$$

for $v\in C_p^{\cap }$ (see Definition 4.1 in Ishii and Smillie 2010). One can then verify that it is invariant and expanding (Proposition 4.3 in Ishii and Smillie 2010) and indeed non-empty (Corollary 4.21 in Ishii and Smillie 2010). This yields the hyperbolicity of the cubic Hénon map in Theorem 5.8 on its Julia set.

Next we prove the non-planarity of the cubic Hénon map $g=f_{a, b}$. Suppose that $g : J_g\rightarrow J_g$ is topologically conjugate to $\hat{p} : \hat{J}_p\rightarrow \hat{J}_p$ for an expanding polynomial p. Then, by the comparison of the entropy, we know that the degree of p is 3. Since g has an attractive 2-cycle and since its Julia set $J_g$ is neither connected nor totally disjoint, we know that one of the two critical points of p escapes to infinity and the other converges to an attractive 2-cycle. This puts certain constraints on $\hat{J}_p$. On the other hand, by using the four Poincaré boxes in the proof of Theorem 5.8, we can analyze the topology of $J_g$ in terms of a symbolic dynamics (Theorem 4.23 in Ishii and Smillie 2010). By comparing the topological types of some path-components of $J_g$ and $\hat{J}_p$, we finally arrive at a contradiction (see the end of Section 4 Ishii 2008 for more details).

6 Three Methods to Describe Hyperbolic Julia Sets

In this section Problems (v), (vi) and (vii) will be discussed. Namely, we construct Hubbard trees, iterated monodromy groups and associated automata for hyperbolic polynomial diffeomorphisms of $\mathbb {C}^2$ starting from the family of Poincaré boxes employed in Sect. 5.2.

6.1 Hubbard Trees

Let us first explain the construction of Hubbard trees (Ishii 2009) by using the non-planar map obtained in Theorem 5.8. Let $\{\mathcal {A}_i\}_{i=0}^3$ be the family of Poincaré boxes appeared in the proof of Theorem 5.8 where $\mathcal {A}_i=A_{x, i}\times A_{y, i}$. Define the forward Julia set of the disjoint system (2) by

$$\begin{aligned} J_+^0(\tilde{f})\equiv \bigcap _{n\ge 0}(\tilde{f}\circ \tilde{\iota }_{\mathcal {A}}^{-1})^{-n}(\widetilde{\mathcal {A}}^0) \end{aligned}$$

(and similar definition for $J_+^1(\tilde{f})$). Since $A_{y, i}$ is simply connected and the disjoint system (2) is a hyperbolic system, one can show that $J_+^m(\tilde{f})\cap \mathcal {A}_i$ forms a lamination where every leaf is a vertical disk of degree one in $\mathcal {A}_i$, i.e. the projection of the disk to $A_{y, i}$ is a proper map of degree one. Moreover, one may assume that these disks are straight vertical (see the comment following Lemma 5.5 of Ishii and Smillie 2010). In particular, the image of any leaf of the lamination by the projection $\pi _x : \mathcal {A}_i\rightarrow A_{x, i}$ is one point. Therefore, by letting $\mathcal {S}_i\equiv A_{x, i}$, $\tilde{\sigma }\equiv \pi _x\circ \tilde{f}\circ \pi _x^{-1}$ and $\tilde{\iota }_{\mathcal {S}}\equiv \pi _x\circ \tilde{\iota }_{\mathcal {A}}\circ \pi _x^{-1}$, we obtain the multivalued dynamical system:

$$\begin{aligned} \tilde{\iota }_{\mathcal {S}}, \tilde{\sigma } \ : \ \widetilde{\mathcal {S}}^1\equiv \bigsqcup _{(i, j)\in \Sigma ^1}\mathcal {S}_i\cap \tilde{\sigma }^{-1}(\mathcal {S}_j)\ \longrightarrow \ \widetilde{\mathcal {S}}^0\equiv \bigsqcup _{j\in \Sigma ^0}\mathcal {S}_j. \end{aligned}$$

(3)

Since the disjoint system (2) expands the horizontal Poincaré cone field, the system (3) equipped with the Poincaré metrics in $\widetilde{\mathcal {S}}^m$ ($m=0, 1$) is expanding in the sense of Definition 1.8.

Now we proceed as explained in item (v) in Sect. 1, but here is a crucial difference which comes from the overlaps of Poincaré boxes. Let $\mathrm {pr}_{\mathcal {A}} : \widetilde{\mathcal {A}}^m\rightarrow \mathcal {A}^m$ be the obvious map.

Definition 6.1

Let $D_0$ and $D_1$ be two leaves in the lamination $J_+^m(\tilde{f})$ ($m=0, 1$). We call the pair $\{D_0, D_1\}$ a pair of pinching disks in $\widetilde{\mathcal {A}}^m$ if $\mathrm {pr}_{\mathcal {A}}(D_0)\cap \mathrm {pr}_{\mathcal {A}}(D_1) \ne \emptyset$. Such disks are called pinching disks in $\widetilde{\mathcal {A}}^m$ (see Fig. 10).

Fig. 10

Intersecting pair of pinching disks $\{D_0, D_1\}$

The images of the pinching disks in $\widetilde{\mathcal {A}}^m$ by the projection $\pi _x$ to $\widetilde{\mathcal {S}}^m$ is called the pinching locus in $\widetilde{\mathcal {S}}^m$ and denoted by $P^m$. We fill up all holes in $\widetilde{\mathcal {S}}^m$ ($m=0, 1$) and choose a point from each hole which we call a center. Let $C^m$ ($m=0, 1$) be the set of centers in $\widetilde{\mathcal {S}}^m$. We define $\mathcal {H}^m$ to be the legal hull of $P^m\cup C^m$ in $\widetilde{\mathcal {S}}^m$ (just as in item (v) of Sect. 1) and then replace every point $c\in \mathcal {H}^m$ which belongs to $C^m$ by a loop to obtain a tree $\widetilde{\mathcal {T}}^m$ decorated with loops. The map $\tilde{\sigma }$ naturally induces a map $\tilde{\tau } : \widetilde{\mathcal {T}}^1\rightarrow \widetilde{\mathcal {T}}^0$ up to homotopy. One can also define a map $\tilde{\iota }_{\mathcal {T}} : \widetilde{\mathcal {T}}^1\rightarrow \widetilde{\mathcal {T}}^0$ up to homotopy which is the identity on $\widetilde{\mathcal {T}}^0$ and smashing each connected component of $\widetilde{\mathcal {T}}^1{\setminus }\widetilde{\mathcal {T}}^0$ to a point. We say that two points t and $t'$ in $P^m$ form a pinching pair and denoted as $t\approx _m t'$ if there exist a pair of pinching disks $\{D, D'\}$ in $\widetilde{\mathcal {A}}^m$ so that $t=\pi _x(D)$ and $t'=\pi _x(D')$.

Definition 6.2

(Ishii 2011, 2014) We call the multivalued dynamical system:

$$\begin{aligned} \tilde{\iota }_{\mathcal {T}}, \tilde{\tau } : \widetilde{\mathcal {T}}^1\longrightarrow \widetilde{\mathcal {T}}^0 \end{aligned}$$

together with the set of pinching pairs in $P^m$ the Hubbard tree for f.

Remark 6.3

The above definition has been first presented in Definition 2.4 of Ishii (2011) which is slightly different from the original one in Definition 4.5 of Ishii (2009).

Now we construct a topological model for the Julia set starting from a Hubbard tree (Ishii 2011, 2014). Consider first the shift map on the orbit space:

$$\begin{aligned} \tilde{\tau }^{\pm \infty } : \widetilde{\mathcal {T}}^{\pm \infty }\longrightarrow \widetilde{\mathcal {T}}^{\pm \infty } \end{aligned}$$

of the Hubbard tree $\tilde{\iota }_{\mathcal {T}}, \tilde{\tau } : \widetilde{\mathcal {T}}^1\rightarrow \widetilde{\mathcal {T}}^0$. For $\underline{t}=(t_n)_{n\in \mathbb {Z}}$ and $\underline{t}'=(t'_n)_{n\in \mathbb {Z}}$ in $\widetilde{\mathcal {T}}^{\pm \infty }$, we define $\underline{t}\approx _{\pm \infty }\underline{t}'$ if either $t_i=t'_i$ or $t_i\approx _1t'_i$ holds for any $i\in \mathbb {Z}$. We also write $\underline{t}\sim _{\pm \infty }\underline{t}'$ if there exist a sequence of points $\underline{t}=\underline{t}^0, \underline{t}^1, \dots , \underline{t}^k=\underline{t}'$ in $\widetilde{\mathcal {T}}^{\pm \infty }$ with $\underline{t}^j\approx _{\pm \infty }\underline{t}^{j+1}$ for all $0\le j\le k-1$. Then, $\sim _{\pm \infty }$ defines an equivalence relation in $\widetilde{\mathcal {T}}^{\pm \infty }$, hence the factor map:

$$\begin{aligned} \tilde{\tau }^{\pm \infty }/_{\sim _{\pm \infty }} : \widetilde{\mathcal {T}}^{\pm \infty }/_{\sim _{\pm \infty }}\longrightarrow \widetilde{\mathcal {T}}^{\pm \infty }/_{\sim _{\pm \infty }} \end{aligned}$$

of $\tilde{\tau }^{\pm \infty }$ is well-defined.⁹

The following theorem states that the dynamics of f is reconstructed from its Hubbard tree.

Theorem 6.4

(Ishii 2011, 2014) Let $\iota _{\mathcal {A}}, f : \mathcal {A}^1\rightarrow \mathcal {A}^0$ be a hyperbolic system and let $\mathfrak {T}$ be its Hubbard tree. If $\mathcal {A}^{\pm \infty }$ is a hyperbolic set and $J_f\subset \mathcal {A}^0$, then $f : J_f\rightarrow J_f$ is topologically conjugate to the factor $\tilde{\tau }^{\pm \infty }/_{\sim _{\pm \infty }} : \widetilde{\mathcal {T}}^{\pm \infty }/_{\sim _{\pm \infty }}\longrightarrow \widetilde{\mathcal {T}}^{\pm \infty }/_{\sim _{\pm \infty }}$.

The proof of this result is based on the method of homotopy shadowing (Ishii and Smillie 2010).

Here we present two examples of hyperbolic systems and their Hubbard trees. The first one called the Basilica-Horseshoe map denoted by $f=f_{\mathrm {BH}}$ consists of three Poincaré boxes which are mapped with each other as described in Fig. 11. This looks similar to the non-planar cubic Hénon map in Fig. 9, but we merge the right-upper boxes $\mathcal {A}_1$ and $\mathcal {A}_2$ in Fig. 9 into one box (and denote it by $\mathcal {A}_2$ in Fig. 11). The map $f_{\mathrm {BH}}$ has a unique attractive cycle of period two and its Julia set is disconnected. Note that the restriction $f_{\mathrm {BH}} : \mathcal {A}_2\cap f^{-1}_{\mathrm {BH}}(\mathcal {A}_1)\rightarrow \mathcal {A}_1$ looks like the Horseshoe map, and the restriction $f_{\mathrm {BH}} : \mathcal {A}_3\cap f^{-1}_{\mathrm {BH}}(\mathcal {A}_1)\rightarrow \mathcal {A}_1$ looks like the Basilica map. Figure 12 describes the Hubbard tree for $f_{\mathrm {BH}}$.

Fig. 11

Poincaré boxes for the Basilica-Horseshoe map

Fig. 12

Hubbard tree for the Basilica-Horseshoe map $f_{\mathrm {BH}}$

Fig. 13

Poincaré boxes for the Airplane-Basilica map

Fig. 14

Hubbard tree for the Airplane-Basilica map $f_{\mathrm {AB}}$

The second one called the Airplane-Basilica map denoted by $f=f_{\mathrm {AB}}$ consists of three Poincaré boxes which are mapped with each other as described in Fig. 13. The map $f_{\mathrm {AB}}$ has two attractive cycles of period two and three respectively, and its Julia set is connected. Note that the map $f_{\mathrm {AB}}$ looks like a mixture of the Airplane map and the Basilica map. Figure 14 describes the Hubbard tree for $f_{\mathrm {AB}}$.

6.2 IMG Actions

In this section we introduce the notion of the iterated monodromy groups for a class of hyperbolic polynomial diffeomorphisms of $\mathbb {C}^2$. Here we formulate an iterated monodromy group as an inverse semigroup action on certain quotient space of the disjoint union of several preimage trees (see Appendix of Ishii 2014 for the generality on inverse semigroup actions). In the original paper Ishii (2014) we defined this notion in terms of its Hubbard tree in an algorithmic way. Here, however, we give an intuitive definition starting from the data of a family of Poincaré boxes with overlaps. This definition is purely geometric and we explain it by the Basilica-Horseshoe map $f_{\mathrm {BH}}$ appeared in the previous section.

Let $A_x$ and $A_y$ be connected open subsets of $\mathbb {C}$ with $A_y$ simply connected and let $\mathcal {A}=A_x\times A_y$ be a Poincaré box. Let $\iota _{\mathcal {A}}, f : \mathcal {A}\cap f^{-1}(\mathcal {A})\rightarrow \mathcal {A}$ be a hyperbolic system obtained as a small perturbation of an expanding polynomial map $p : A_x\cap p^{-1}(A_x)\rightarrow A_x$ of degree $d\ge 2$. By an appropriate change of coordinates (see Subsection 5.1 of Ishii and Smillie 2010) we have $p\circ \mathrm {pr}_x=\mathrm {pr}_x\circ f$, where $\mathrm {pr}_x : \mathcal {A}\rightarrow A_x$ is the x-projection. Then, for any vertical disk V in $\mathcal {A}$, the slice $V\cap f^n(\mathcal {A})$ has a nested structure so that $V\cap J^0_-(\tilde{f})$ becomes a Cantor set, where

$$\begin{aligned} J_-^0(f)\equiv \bigcap _{n\ge 0}(f\circ \iota _{\mathcal {A}}^{-1})^n(\mathcal {A}). \end{aligned}$$

Hence, the projection $\mathrm {pr}_x : J^0_-(\tilde{f})\rightarrow A_x$ is a fibration with Cantor fibers, and the Cantor fiber over $x\in A_x$ can be identified with the “ideal boundary” of the preimage tree of p rooted at x. With this identification, the action of the iterated monodromy group of p on T is interpreted as the holonomy group action on a Cantor fiber. Moreover, we note that each connected component of $V\cap f^n(\mathcal {A})$ can be identified with certain subtree of T.

Next consider the disjoint model $\tilde{\iota }_{\mathcal {A}}, \tilde{f} : \widetilde{\mathcal {A}}^1\rightarrow \widetilde{\mathcal {A}}^0$ of a hyperbolic system $\iota _{\mathcal {A}},f : \mathcal {A}^1\rightarrow \mathcal {A}^0$ with several Poincaré boxes, where $\widetilde{\mathcal {A}}^0\equiv \bigsqcup _{i\in \Sigma ^0}\mathcal {A}_i$ and $\mathcal {A}^0\equiv \bigcup _{i\in \Sigma ^0}\mathcal {A}_i$ (see Sect. 5.2 for the definition of $\widetilde{\mathcal {A}}^1$ and $\mathcal {A}^1$). Then, the lamination in each box $J^0_-(\tilde{f})\cap \mathcal {A}_i$ looks similar to the previous case, where

$$\begin{aligned} J_-^0(\tilde{f})\equiv \bigcap _{n\ge 0}(\tilde{f}\circ \tilde{\iota }_{\mathcal {A}}^{-1})^n(\widetilde{\mathcal {A}}^0) \end{aligned}$$

(and similar definition for $J_-^1(\tilde{f})$). To the lamination in $J^0_-(\tilde{f})\cap \mathcal {A}_i$ we can associate a preimage tree $T_i$. Note that since the degree of the map varies from boxes to boxes, the preimage trees are no more regular but “SFT-like”.

Finally, we define the iterated monodromy group for a hyperbolic system $\iota _{\mathcal {A}}, f : \mathcal {A}^1\rightarrow \mathcal {A}^0$ with overlapping Poincaré boxes $\mathcal {A}^0\equiv \bigcup _{i\in \Sigma ^0}\mathcal {A}_i$ and $\mathcal {A}^1=\bigcup _{(i, j)\in \Sigma ^1}\mathcal {A}_i\cap f^{-1}(\mathcal {A}_j)$. To do this, we first need to define a space on which the iterated monodromy group acts. Note that in this case the laminations in $\mathrm {pr}_{\mathcal {A}}(J^m_-(\tilde{f})\cap \mathcal {A}_i)$ are overlapping with each other. Since a geometric interpretation of an iterated monodromy group is the holonomy group action for the laminations, we need to understand how these laminations are continued from boxes to boxes. Each section of the lamination is identified with the ideal boundary of the corresponding preimage tree, so the overlapping data should be described by certain identification between the preimage trees.

Below we explain this identification rule for the Basilica-Horseshoe map. In this process, the following principle is important which extends the observation for the case of a single Poincaré box; there is a one-to-one correspondence between a subtree $T_{i_N}$ in $T_{i_0}$ of the form:

$$\begin{aligned} b_{i_0}\longleftarrow b'_{i_1}\longleftarrow \ \cdots \ \longleftarrow b^{(N-1)}_{i_{N-1}}\leftarrow T_{i_N} \end{aligned}$$

and a connected component of

$$\begin{aligned} V_{b_{i_0}}\cap \tilde{f}(\mathcal {A}_{i_1})\cap \cdots \cap \tilde{f}^{N-1}(\mathcal {A}_{i_{N-1}})\cap \tilde{f}^N(\mathcal {A}_{i_N}), \end{aligned}$$

where $b_i$ is the base-point of a connected component $\mathcal {T}_i$ of $\mathcal {T}^0$ (hence it is the root of the preimage tree $T_i$) and $V_{b_i}=\mathrm {pr}_x^{-1}(b_i)$ is the vertical disk in $\mathcal {A}_i$ over $b_i$.

First note that the transitions between the Poincaré boxes for this map are described in Fig. 9 and hence its transition diagram is given in Fig. 15.

Step 1: $T_2$ can be seen as a subtree of $T_1$. The two subtrees $T'_2$’s in the first level of $T_1$ (the zeroth level is the root) correspond to the two connected components of $V_{b_1}\cap \tilde{f}(\mathcal {A}_2)$ denoted by $D'_{b_1}$ and $E'_{b_1}$ respectively. Similarly, the subtree $T'_2$ in the first level of $T_2$ corresponds to $D'_{b_2}=V_{b_2}\cap \tilde{f}(\mathcal {A}_2)$. Then, one of $\mathrm {pr}_{\mathcal {A}}(D'_{b_1})$ and $\mathrm {pr}_{\mathcal {A}}(E'_{b_1})$ coincides with $\mathrm {pr}_{\mathcal {A}}(D'_{b_2})$ in $\mathcal {A}$ by letting $\mathrm {pr}_{\mathcal {A}}(V_{b_1})\cap \mathrm {pr}_{\mathcal {A}}(V_{b_2})\ne \emptyset$. Assume that $D'_{b_1}$ is such a connected component and identify the corresponding subtree $T'_2$ in the first level of $T_1$ with the subtree $T'_2$ in the first level of $T_2$. Similarly we identify one of two $T'_3$’s in the first level of $T_1$ with $T'_3$ in the first level of $T_2$. This gives a bijective correspondence between the infinite paths ending at $b_1$ in $T_1$ through the union of $T'_2$ and $T'_3$ in the first level of $T_1$ and the ones ending at $b_2$ in $T_2$. Hence $T_2$ can be seen as a subtree of $T_1$ (see Fig. 16).

Fig. 15

Transition diagram for the Basilica-Horseshoe map

Fig. 16

Identifying the preimage trees for the Basilica-Horseshoe map

Step 2: $T_3$ can be seen as a subtree of $T_1$. Let $T'_2$ (resp. $T'_3$) in the first level of $T_1$ be the one corresponding to the remaining component of $V_{b_1}\cap \tilde{f}(\mathcal {A}_2)$ (resp. $V_{b_1}\cap \tilde{f}(\mathcal {A}_3)$). We identify these two subtrees with $T_3$. To the root $b'_2$ of this subtree $T'_2$ in $T_1$ we have two edges from the subtrees $T''_2$ and $T''_3$ in the second level of $T_1$, and to the root $b'_3$ of the subtree $T'_3$ in the first level of $T_1$ we have an edge from $T''_1$ in the second level of $T_1$. To the root $b'_1$ of the subtree $T'_1$ in the first level of $T_3$ we have four edges from two $T''_2$’s and two $T''_3$’s in the second level of $T_3$.

The two subtrees $T''_2$’s in the second level of $T_3$ correspond to the two connected components of $V_{b_3}\cap \tilde{f}(\mathcal {A}_1)\cap \tilde{f}^2(\mathcal {A}_2)$ denoted by $D''_{b_3}$ and $E''_{b_3}$ respectively. Similarly, the subtree $T''_2$ in the second level of $T_1$ corresponds to $E''_{b_1}=V_{b_1}\cap \tilde{f}(\mathcal {A}_2)\cap \tilde{f}^2(\mathcal {A}_2)$. Then, one of $\mathrm {pr}_{\mathcal {A}}(D''_{b_3})$ and $\mathrm {pr}_{\mathcal {A}}(E''_{b_3})$ coincides with $\mathrm {pr}_{\mathcal {A}}(E''_{b_1})$ by letting $\mathrm {pr}_{\mathcal {A}}(V_{b_1})\cap \mathrm {pr}_{\mathcal {A}}(V_{b_3})\ne \emptyset$. Assume that $D''_{b_3}$ is such a connected component and identify the corresponding subtree $T''_2$ in the second level of $T_3$ with the subtree $T''_2$ in the second level of $T_1$. Similarly we identify one of two $T_3''$’s in the second level of $T_3$ with $T_3''$ in the second level of $T_1$.

Now we repeat this procedure. Then, for most infinite paths in $T_1$ ending at the root point one can find corresponding paths which are eventually identified by the procedure above. The only exception is the path:

$$\begin{aligned} b_1\longleftarrow b'_3\longleftarrow b''_1\longleftarrow b'''_3\longleftarrow \ \cdots \end{aligned}$$

in $T_1$. We define its corresponding path in $T_3$ to be the one:

$$\begin{aligned} b_3\longleftarrow b'_1\longleftarrow b''_3\longleftarrow b'''_1\longleftarrow \ \cdots . \end{aligned}$$

This gives a bijective correspondence between the infinite paths ending at $b_1$ in $T_1$ through the union of $T'_2$ and $T'_3$ in the first level of $T_1$ and the ones ending at $b_3$ in $T_3$. Hence $T_3$ can be seen as a subtree of $T_1$ (see Fig. 16 again).

Step 3: Action of $\mathrm {IMG}(\mathfrak {T})$ on T. The discussion in the previous step defines an equivalence relation called the holonomy equivalence denoted by $\sim _{\mathrm {holo}}$ in the disjoint union $T_1\sqcup T_2\sqcup T_3$ of the preimage trees. Set

$$\begin{aligned} T\equiv \big (T_1\sqcup T_2\sqcup T_3\big )/_{\sim _{\mathrm {holo}}}. \end{aligned}$$

In the example discussed above, we identified $T_2$ and $T_3$ as subtrees of $T_1$, therefore $T=T_1$.

Let $\mathrm {IMG}(\widetilde{\mathfrak {T}})_i$ be the iterated monodromy group associated to a connected component $\mathcal {T}_i$ of $\mathcal {T}^0$ acting on $T_i$. Let $\kappa : T_1\sqcup T_2\sqcup T_3\rightarrow T$ be the natural projection. Then, the action of $\mathrm {IMG}(\widetilde{\mathfrak {T}})_i$ on $T_i$ descends to an inverse semigroup action on T which is denoted by $\kappa _{*}(\mathrm {IMG}(\widetilde{\mathfrak {T}})_i)$. We define

$$\begin{aligned} \mathrm {IMG}(\mathfrak {T}) \equiv \kappa _{*}(\mathrm {IMG}(\widetilde{\mathfrak {T}})_1) \odot \kappa _{*}(\mathrm {IMG}(\widetilde{\mathfrak {T}})_2) \odot \kappa _{*}(\mathrm {IMG}(\widetilde{\mathfrak {T}})_3), \end{aligned}$$

where $G_1\odot G_2$ denotes the free product of two inverse semigroup actions $G_1$ and $G_2$ (see Section 7 of Ishii 2014), and call it the iterated monodromy group of the Hubbard tree $\mathfrak {T}$.

Let $\Pi$ be the set of arrows in the diagram of Fig. 15 and put

$$\begin{aligned} \Pi ^{-\infty }\equiv \Pi ^{-\infty }_1\sqcup \Pi ^{-\infty }_2\sqcup \Pi ^{-\infty }_3, \end{aligned}$$

where

$$\begin{aligned} \Pi ^{-\infty }_i\equiv \big \{(\pi _n)_{n\le -1}\in \Pi ^{-\mathbb {N}} : h(\pi _{n-1})=t(\pi _n) \text{ for } \text{ all } n\le -1 \text{ and } h(\pi _{-1})=i \big \} \end{aligned}$$

is the space of symbol sequences corresponding to the infinite paths in $T_i$ ending at its root point. Hence, $\Pi ^{-\infty }$ can be seen as the “ideal boundary” of T. Note that every element in $\Pi ^{-\infty }$ corresponds to an infinite path in T ending at some root point which we call a rooted path.

Let $\Pi ^{\pm \infty }$ be the set of bi-infinite paths in the diagram of Fig. 15.

Definition 6.5

We say that $\underline{\pi }=(\pi _n)_{n\in \mathbb {Z}}$ and $\underline{\pi }'=(\pi '_n)_{n\in \mathbb {Z}}$ in $\Pi ^{\pm \infty }$ are holonomy equivalent and write $\underline{\pi }\sim _{\mathrm {holo}}\underline{\pi }'$ if for any $n\in \mathbb {Z}$, one of the following two conditions is satisfied;

• (i) there exists $N\le n$ so that the subpaths corresponding to $\cdots \pi _{N-1}\pi _N$ and $\cdots \pi '_{N-1}\pi '_N$ of the rooted paths corresponding to $\cdots \pi _{n-1}\pi _n$ and $\cdots \pi '_{n-1}\pi '_n$ in T respectively are identified in $T/_{\sim _{\mathrm {holo}}}$, or

• (ii) the sequences $\cdots \pi _{n-1}\pi _n$ and $\cdots \pi '_{n-1}\pi '_n$ form an exceptional pair.

The action of the iterated monodromy group $\mathrm {IMG}(\mathfrak {T})$ on $T^{*}/_{{\sim }_{\mathrm {holo}}}$ induces an action on $\Pi ^{*}/_{{\sim }_{\mathrm {holo}}}$ through the bijection $\widetilde{\Lambda } : \Pi ^{*}\rightarrow T^{*}$ which extends to an action on $\Pi ^{-\infty }/_{{\sim }_{\mathrm {holo}}}$. We call it the standard action of $\mathrm {IMG}(\mathfrak {T})$ on $\Pi ^{-\infty }/_{{\sim }_{\mathrm {holo}}}$. For $e, e'\in \Pi ^{\pm \infty }/_{{\sim }_{\mathrm {holo}}}$ we write $e\approx _{\mathrm {asym}} e'$ if $\underline{\pi }\sim _{\mathrm {asym}}\underline{\pi }$ in the extended sense of Definition 1.17 (see Subsection 9.4 of Bartholdi et al. 2003 for more precise definition) for some representatives $\underline{\pi }\in e$ and $\underline{\pi }'\in e'$. The equivalence relation generated by $\approx _{\mathrm {asym}}$ is called the asymptotic equivalence in $\Pi ^{\pm \infty }/_{{\sim }_{\mathrm {holo}}}$ and is denoted again by $\approx _{\mathrm {asym}}$. Similarly, for $E, E'\in \Pi ^{\pm \infty }/_{\sim _{\mathrm {asym}}}$ we write $E\approx _{\mathrm {holo}} E'$ if $\underline{\pi }\sim _{\mathrm {holo}} \underline{\pi }'$ in the sense of Definition 6.5 for some representatives $\underline{\pi }\in E$ and $\underline{\pi }'\in E'$. The equivalence relation generated by $\approx _{\mathrm {holo}}$ is called the holonomy equivalence in $\Pi ^{\pm \infty }/_{{\sim }_{\mathrm {asym}}}$ and is denoted again by $\approx _{\mathrm {holo}}$.

Definition 6.6

We call

$$\begin{aligned} \mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}\equiv \bigl (\Pi ^{\pm \infty }/_{{\sim }_{\mathrm {holo}}}\bigr )/_{\approx _{\mathrm {asym}}} \end{aligned}$$

the limit solenoid of the iterated monodromy group $\mathrm {IMG}(\mathfrak {T})$ for the Hubbard tree $\mathfrak {T}$.

We denote by $s : \mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}\rightarrow \mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}$ the factor of the shift map on $\Pi ^{\pm \infty }$.

Theorem 6.7

(Ishii 2014) Let $\iota _{\mathcal {A}}, f : \mathcal {A}^1\rightarrow \mathcal {A}^0$ be a hyperbolic system and let $\mathfrak {T}$ be its Hubbard tree. If $\mathcal {A}^{\pm \infty }$ is a hyperbolic set and $J_f\subset \mathcal {A}^0$, then $f : J_f\rightarrow J_f$ is topologically conjugate to the factor $s : \mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}\rightarrow \mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}$ of the shift map.

The proof of this result is based on the method of homotopy shadowing (Ishii and Smillie 2010).

6.3 Automata

In his thesis Oliva (1998) introduced an algorithm for drawing gradient lines which allows the gradient lines to be drawn all the way until they land at $J_f$. Thus the computer can determine whether two gradient lines land at the same point. These pictures are colored in a way that shows the solenoidal coding, and conversely, given a point of the solenoid, the computer can plot the corresponding ray. This gave Oliva the data (pairs of identified solenoidal points) which were the basis for his automaton. Figures 4.16, 4.17 and 4.18 in Oliva (1998) give plots where many pairs of rays were tested to corroborate his automaton.

The purpose of this section is to construct automata associated with iterated monodromy group actions for certain hyperbolic polynomial diffeomorphisms of $\mathbb {C}^2$ along with Ishii (2014) (see also Lemma 1 in Fried 1987 for a related result). The idea of the construction is as follows. First, the inverse semigroup actions of $\mathrm {IMG}(\widetilde{\mathfrak {T}})_i$ give rise to the direct sum of the inverse semigroup actions of

$$\begin{aligned} \mathrm {IMG}(\widetilde{\mathfrak {T}})\equiv \coprod _{i\in \Sigma ^0}\mathrm {IMG}(\widetilde{\mathfrak {T}})_i \end{aligned}$$

on the disjoint union of preimage trees $\bigsqcup _{i\in \Sigma ^0}T_i$ (see Appendix of Ishii 2014 for the definition of the direct sum of inverse semigroup actions). We next reformulate the notion of holonomy equivalence in terms of an inverse semigroup action called the holonomy pinching group $\mathcal {I}_{\mathfrak {T}}$. Intuitively the action of $\mathrm {IMG}(\widetilde{\mathfrak {T}})$ describes the dynamics in the expanding direction and the action of $\mathcal {I}_{\mathfrak {T}}$ describes the dynamics in the contracting direction. It is shown that the two actions are both self-similar and hence can define corresponding automata. We define the automaton for a polynomial diffeomorphisms of $\mathbb {C}^2$ as certain power of the product of the two automata (Definition 6.12) and show that the quotient space with respect to the equivalence relation generated by the automaton is identical to $\mathcal {S}_{\mathrm {IMG}(\mathfrak {T})}$ (Theorem 6.13).

To accomplish this procedure we first reformulate the notion of an automaton as follows and introduce the notion of the product of two automata. Let $\Pi$ be a set called an alphabet.

Definition 6.8

An automaton over $\Pi$ is a triple $\mathfrak {A}=(Q, q, \pi )$, where (i) Q is a set, and (ii) $\pi : \Pi \times Q\dashrightarrow \Pi$ and $q : \Pi \times Q\dashrightarrow Q$ are partially defined maps with a common domain.

It is often convenient to write an automaton $\mathfrak {A}=(Q, q, \pi )$ as a pair of partially defined maps $\tau =(q, \pi ) : \Pi \times Q \dashrightarrow Q\times \Pi$. Given two automata $\mathfrak {A}=(Q_{\mathfrak {A}}, q_{\mathfrak {A}}, \pi _{\mathfrak {A}})$ and $\mathfrak {B}=(Q_{\mathfrak {B}}, q_{\mathfrak {B}}, \pi _{\mathfrak {B}})$ over a same alphabet $\Pi$ we define their product $\mathfrak {A}\mathfrak {B}=(Q_{\mathfrak {A}}\times Q_{\mathfrak {B}}, q_{\mathfrak {A}\mathfrak {B}}, \pi _{\mathfrak {A}\mathfrak {B}})$ over $\Pi$, where $\tau _{\mathfrak {A}\mathfrak {B}}=(q_{\mathfrak {A}\mathfrak {B}}, \pi _{\mathfrak {A}\mathfrak {B}}) : \Pi \times Q_{\mathfrak {A}}\times Q_{\mathfrak {B}}\dashrightarrow Q_{\mathfrak {A}}\times Q_{\mathfrak {B}}\times \Pi$ is given by the successive compositions:

$$\begin{aligned} \Pi \times Q_{\mathfrak {A}}\times Q_{\mathfrak {B}}\dashrightarrow Q_{\mathfrak {A}}\times \Pi \times Q_{\mathfrak {B}}\dashrightarrow Q_{\mathfrak {A}}\times Q_{\mathfrak {B}}\times \Pi \end{aligned}$$

of first $\tau _{\mathfrak {A}}=(q_{\mathfrak {A}}, \pi _{\mathfrak {A}})$ and next $\tau _{\mathfrak {B}}=(q_{\mathfrak {B}}, \pi _{\mathfrak {B}})$.

An automaton $\mathfrak {A}=(Q, q, \pi )$ over $\Pi$ defines a binary relation $R_{\mathfrak {A}}\subset \Pi ^{\mathbb {Z}}\times \Pi ^{\mathbb {Z}}$. Let $\underline{\pi }=(\pi _n)_{n\in \mathbb {Z}}, \underline{\pi }'=(\pi _n')_{n\in \mathbb {Z}}\in \Pi ^{\mathbb {Z}}$ be two bi-infinite sequences. We say that $(\underline{\pi }, \underline{\pi }')\in \Pi ^{\mathbb {Z}}\times \Pi ^{\mathbb {Z}}$ belongs to $R_{\mathfrak {A}}$ if there exists a sequence $(q_n)_{n\in \mathbb {Z}}\in Q^{\mathbb {Z}}$ so that $q_{n+1}=q(\pi _n, q_n)$ and $\pi '_n=\pi (\pi _n, q_n)$ hold.

From an automaton $\mathfrak {A}=(Q, q, \pi )$ over an alphabet $\Pi$, one can construct a doubly labeled directed graph called the Moore diagram of $\mathfrak {A}$. Its vertex set is given by Q, and for $(\pi ', q')\in \Pi \times Q$ in the common domain of $\pi$ and q we draw an arrow from $q(\pi ', q')$ to $q'$ labeled by the pair $\pi ' | \pi (\pi ', q')$. Note that the direction of arrows defined here is opposite to the one in Bartholdi et al. (2003).

Definition 6.9

Let $\mathfrak {A}$ be an automaton over $\Pi$. For $\underline{\pi }=(\pi _n)_{n\in \mathbb {Z}}, \underline{\pi }'=(\pi '_n)_{n\in \mathbb {Z}}\in \Pi ^{\mathbb {Z}}$ we write $\underline{\pi }\sim _{\mathfrak {A}}\underline{\pi }$ if there exists a bi-infinite path in the Moore diagram of $\mathfrak {A}$ along which the sequence of labelings is $(\pi _n | \pi '_n)_{n\in \mathbb {Z}}$.

This definition is consistent with the binary relation $R_{\mathfrak {A}}$. Namely, for $\underline{\pi }, \underline{\pi }'\in \Pi ^{\mathbb {Z}}$ we have $(\underline{\pi }, \underline{\pi }')\in R_{\mathfrak {A}}$ if and only if $\underline{\pi } \sim _{\mathfrak {A}} \underline{\pi }'$. When the binary relation $R_{\mathfrak {A}}$ is an equivalence relation, we say that $\mathfrak {A}$ generates the equivalence relation $\sim _{\mathfrak {A}}$.

Given two binary relations $R_1, R_2\subset \Pi ^{\mathbb {Z}}\times \Pi ^{\mathbb {Z}}$ in $\Pi ^{\mathbb {Z}}$, their product $R_1R_2\subset \Pi ^{\mathbb {Z}}\times \Pi ^{\mathbb {Z}}$ is defined as follows; we say $(\underline{\pi }^1, \underline{\pi }^2)\in R_1R_2$ iff there exists $\underline{\delta }\in \Pi ^{\mathbb {Z}}$ so that $(\underline{\pi }^1, \underline{\delta })\in R_1$ and $(\underline{\delta }, \underline{\pi }^2)\in R_2$ hold. Similarly, given two equivalence relations $\sim _1, \sim _2\, \subset \Pi ^{\mathbb {Z}}\times \Pi ^{\mathbb {Z}}$ in $\Pi ^{\mathbb {Z}}$, their product $\sim _1\sim _2$ is defined as the transitive closure of $\sim _1$ and $\sim _2$. In particular, we have

$$\begin{aligned} \sim _\mathfrak {A}\sim _\mathfrak {B}\ =\bigcup _{m>0}(R_\mathfrak {A}R_\mathfrak {A})^m \end{aligned}$$

if two automata $\mathfrak {A}$ and $\mathfrak {B}$ generate the equivalence relations $\sim _\mathfrak {A}$ and $\sim _\mathfrak {B}$ respectively.

A key property in the construction of finite automata for Hénon maps is

Definition 6.10

Let $M\ge 1$ and let $R_{\mathfrak {A}}$ and $R_{\mathfrak {B}}$ be the binary relations in $\Pi ^{\mathbb {Z}}$ defined by automata $\mathfrak {A}$ and $\mathfrak {B}$ respectively. We say that the product $\mathfrak {A}\mathfrak {B}$ is M -boundedly generating in $\Pi ^{\mathbb {Z}}$ if $R_{(\mathfrak {A}\mathfrak {B})^m} \subset R_{(\mathfrak {A}\mathfrak {B})^M}$ holds for every $m\ge 1$.

Next we reformulate the holonomy equivalence in terms of certain inverse semigroup action. Choose $i, i'\in \Sigma ^0$ with $i\ne i'$ so that $E(i, i')\ne \emptyset$ holds, i.e. $\mathcal {T}_i$ and $\mathcal {T}_{i'}$ are identified at some points by $\approx _{\mathfrak {L}^0}$. For $\cdots \pi _{-2}\pi _{-1}\in \Pi ^{-\infty }_i$ and $\cdots \pi '_{-2}\pi '_{-1}\in \Pi ^{-\infty }_{i'}$, we set $\iota _{\{i, i'\}}(\cdots \pi _{-2}\pi _{-1})\equiv \cdots \pi '_{-2}\pi '_{-1}$ and $\iota _{\{i, i'\}}(\cdots \pi '_{-2}\pi '_{-1})\equiv \cdots \pi _{-2}\pi _{-1}$ if either

• (i) there exists $N\le -1$ so that the subpaths $\cdots \pi _{N-1}\pi _{N}$ and $\cdots \pi '_{N-1}\pi '_{N}$ of the paths $\cdots \pi _{-2}\pi _{-1}$ and $\cdots \pi '_{-2}\pi '_{-1}$ respectively are identified in $T/_{\sim _{\mathrm {holo}}}$, or

• (ii) $\cdots \pi _{-2}\pi _{-1}$ and $\cdots \pi '_{-2}\pi '_{-1}$ form an exceptional pair.

This gives an involution $\iota _{\{i, i'\}}$ from a subset of $\Pi ^{-\infty }_i\sqcup \Pi ^{-\infty }_{i'}$ to itself.

Definition 6.11

The inverse semigroup generated by the maps $\iota _{\{i, i'\}}$ and the identity map on $\Pi ^{-\infty }\equiv \bigsqcup _{i\in \Sigma ^0}\Pi ^{-\infty }_i$ is called the holonomy pinching group of the Hubbard tree $\mathfrak {T}$ and is denoted by $\mathcal {I}_{\mathfrak {T}}$.

The holonomy pinching group $\mathcal {I}_{\mathfrak {T}}$ acts on $\Pi ^{-\infty }$ faithfully and we denote its action of $\iota \in \mathcal {I}_{\mathfrak {T}}$ as $(\cdots \pi _{-2}\pi _{-1})^{\iota }$ for $\cdots \pi _{-2}\pi _{-1}\in \Pi ^{-\infty }$. Note that $\mathcal {I}_{\mathfrak {T}}$ is similar to the holonomy pseudogroup in the foliation theory (Candel and Conlon 2000; Nekrashevych 2006). The holonomy equivalence relation in Definition 6.5 can be then expressed in terms of $\mathcal {I}_{\mathfrak {T}}$ as follows. Let $\underline{\pi }=(\pi _n)_{n\in \mathbb {Z}}, \underline{\pi }'=(\pi _n')_{n\in \mathbb {Z}}\in \Pi ^{\pm \infty }$. Then, $\underline{\pi }\sim _{\mathrm {holo}}\underline{\pi }'$ holds iff for any $n\in \mathbb {Z}$ there exists $\iota _n\in \mathcal {I}_{\mathfrak {T}}$ so that $(\cdots \pi _{n-1}\pi _n)^{\iota _n}=\cdots \pi _{n-1}'\pi _n'$ holds.

The notion of self-similarity of a group action is generalized to the setting of inverse semigroup actions in an appropriate way (see Definition 3.6 of Bartholdi et al. 2003). One can show that the inverse semigroup actions of $\mathcal {I}_{\mathfrak {T}}$ and $\mathrm {IMG}(\widetilde{\mathfrak {T}})$ on $\Pi ^{-\infty }$ are both self-similar and contracting. Let $\mathfrak {A}_{\mathrm {IMG}(\widetilde{\mathfrak {T}})}$ be the Moore diagram of the automaton $(\mathcal {N}_{\mathrm {IMG}(\widetilde{\mathfrak {T}})}, \pi _{\mathrm {IMG}(\widetilde{\mathfrak {T}})}, q_{\mathrm {IMG}(\widetilde{\mathfrak {T}})})$. Also, we denote by $\mathfrak {A}_{\mathcal {I}_{\mathfrak {T}}}$ the Moore diagram of the automaton $(\mathcal {N}_{\mathcal {I}_{\mathfrak {T}}}, \pi _{\mathcal {I}_{\mathfrak {T}}}, q_{\mathcal {I}_{\mathfrak {T}}})$. Denote by $\mathrm {card}(\Sigma ^0)$ the cardinality of $\Sigma ^0$. A key observation is that the product $\mathfrak {A}_{\mathrm {IMG}(\widetilde{\mathfrak {T}})}\mathfrak {A}_{\mathcal {I}_{\mathfrak {T}}}$ is $\mathrm {card}(\Sigma ^0)$-boundedly generating in $\Pi ^{\pm \infty }$ (Proposition 5.17 in Ishii 2014). This motivates to define

Definition 6.12

The $\mathrm {card}(\Sigma ^0)$-th power $(\mathfrak {A}_{\mathrm {IMG}(\widetilde{\mathfrak {T}})}\mathfrak {A}_{\mathcal {I}_{\mathfrak {T}}})^{\mathrm {card}(\Sigma ^0)}$ of the product $\mathfrak {A}_{\mathrm {IMG}(\widetilde{\mathfrak {T}})}\mathfrak {A}_{\mathcal {I}_{\mathfrak {T}}}$ is called the automaton for a Hubbard tree $\mathfrak {T}$ and denoted by $\mathfrak {A}_{\mathfrak {T}}$.

Let $\mathfrak {T}$ be a Hubbard tree and $\Pi ^{\pm \infty }$ be the subshift of finite type associated to it. Let $\mathfrak {A}_{\mathfrak {T}}$ be the automaton for $\mathfrak {T}$. Recall that for $\underline{\pi }=(\pi _n)_{n\in \mathbb {Z}}, \underline{\pi }'=(\pi '_n)_{n\in \mathbb {Z}}\in \Pi ^{\pm \infty }$, we write $\underline{\pi }\sim _{\mathfrak {A}_{\mathfrak {T}}}\underline{\pi }'$ if there exists a bi-infinite path in $\mathfrak {A}_{\mathfrak {T}}$ so that the sequence of labelings along the path is $(\pi _n | \pi '_n)_{n\in \mathbb {Z}}$.

Theorem 6.13

(Ishii 2014) Let $\iota _{\mathcal {A}}, f : \mathcal {A}^1\rightarrow \mathcal {A}^0$ be a hyperbolic system and let $\mathfrak {T}$ be its Hubbard tree. If $\mathcal {A}^{\pm \infty }$ is a hyperbolic set and $J_f\subset \mathcal {A}^0$, then $f : J_f\rightarrow J_f$ is topologically conjugate to the factor $\sigma /_{\sim _{\mathfrak {A}_{\mathfrak {T}}}} : \Pi ^{\pm \infty }/_{\sim _{\mathfrak {A}_{\mathfrak {T}}}}\rightarrow \Pi ^{\pm \infty }/_{\sim _{\mathfrak {A}_{\mathfrak {T}}}}$ of the shift map.

On the other hand, no tight automata theory for Hénon maps is established yet.

7 Exploring the Parameter Space of the Hénon Family

In this section we investigate the complex quadratic Hénon family:

$$\begin{aligned} f_{c, b} : (x, y)\longmapsto (x^2+c-by, x) \end{aligned}$$

where $(c, b)\in \mathbb {C}\times \mathbb {C}^{\times }$. Let us call $\mathbb {C}\times \mathbb {C}^{\times }$ the parameter space of the complex Hénon family. Currently it is a far reaching problem to establish a dynamics-parameter correspondence for such complex 2-dimensional dynamical systems. However, there is a series of interesting conjectures which could be a hint towards this problem. In Sects. 7.1 and 7.2 we explain these conjectures which are based on numerical experiments.

7.1 Is $\rho$ Surjective?

Write $\Sigma _2\equiv \{A, B\}$ and denote by

$$\begin{aligned} \Sigma _2^{\mathbb {Z}}\equiv \big \{\cdots \varepsilon _{-1}\cdot \varepsilon _0\varepsilon _1\cdots : \varepsilon _i\in \Sigma _2\big \} \end{aligned}$$

the space of all two-sided symbol sequences over $\Sigma _2$. We also consider the shift map $\sigma : \Sigma _2^{\mathbb {Z}}\rightarrow \Sigma _2^{\mathbb {Z}}$ given by $\sigma (\cdots \varepsilon _{-1}\cdot \varepsilon _0\varepsilon _1\cdots )\equiv \cdots \varepsilon _{-1}\varepsilon _0\cdot \varepsilon _1\cdots$.

We say that a complex Hénon map is a hyperbolic horseshoe on $\mathbb {C}^2$ if its Julia set $J_{c, b}$ is a hyperbolic set and $f_{c, b} : J_{c, b}\rightarrow J_{c, b}$ is topologically conjugate to the shift map $\sigma : \Sigma _2^{\mathbb {Z}}\rightarrow \Sigma _2^{\mathbb {Z}}$, where $\Sigma _2^{\mathbb {Z}}$ is the space of bi-infinite symbol sequences with two symbols. The complex hyperbolic horseshoe locus is defined as

$$\begin{aligned} \mathcal {H}_{\mathbb {C}}\equiv \big \{(c, b)\in \mathbb {C}\times \mathbb {C}^{\times } : f_{c, b} \text{ is } \text{ a } \text{ hyperbolic } \text{ horseshoe } \text{ on } \mathbb {C}^2 \bigr \} \end{aligned}$$

Note that we do not know if $\mathcal {H}_{\mathbb {C}}$ is connected. We define the shift locus $\mathcal {S}_{c, b}$ for the complex Hénon family as the connected component of $\mathcal {H}_{\mathbb {C}}$ containing the region $\mathcal {H}_{\mathrm {OV}}\equiv \{(c, b)\in \mathbb {C}\times \mathbb {C}^{\times } : |c|>2(1+|b|)^2\}$ found in Oberste-Vorth (1987) (see (1) of Corollary 5.7).

Fix $(c_0, b_0)\in \mathcal {H}_{\mathrm {OV}}$. As in the case of polynomial maps in one complex variable (see item (viii) in Sect. 1), we have an anti-homomorphism:

$$\begin{aligned} \rho : \pi _1(\mathcal {S}_{c, b}, (c_0, b_0))\longrightarrow \mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma ) \end{aligned}$$

satisfying $\rho (\gamma _1\cdot \gamma _2)=\rho (\gamma _2)\rho (\gamma _1)$, where $\mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma )$ is the group of homeomorphisms $\tau : \Sigma _2^{\mathbb {Z}}\rightarrow \Sigma _2^{\mathbb {Z}}$ which commutes with the shift map $\sigma$ on $\Sigma _2^{\mathbb {Z}}$. We call $\rho$ the monodromy presentation of the fundamental group $\pi _1(\mathcal {S}_{c, b}, (c_0, b_0))$.

Conjecture 7.1

(Hubbard, see Bedford and Smillie 2006) The image $\rho (\pi _1(\mathcal {S}_{c, b}, (c_0, b_0)))$ together with the shift map $\sigma : \Sigma _2^{\mathbb {Z}}\rightarrow \Sigma _2^{\mathbb {Z}}$ generate $\mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma )$.

It is easy to see that the locus $\mathcal {H}_{\mathbb {C}}$ is not simply connected; take a loop $\gamma (t)=(c(t), b_0)\in \mathcal {H}_{\mathrm {OV}}$ where c(t) is a large loop surrounding the Mandelbrot set once with $c(0)=c_0$, then $\rho (\gamma )$ exchanges the two symbols A and B. Moreover, Arai (2016) found an element $\gamma \in \pi _1(\mathcal {S}_{c, b}, (c_0, b_0))$ so that $\rho (\gamma )$ has infinite order.

One of the reasons why Conjecture 7.1 seems much more difficult to prove than Theorem 1.22 is that the group $\mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma )$ is “huge” compared to $\mathrm {Aut}(\Sigma _2^{\mathbb {N}_0}, \sigma )$. For example, it is known that $\mathrm {Aut}(\Sigma _2^{\mathbb {Z}}, \sigma )$ contains every finite group and the direct sum of countably many copies of $\mathbb {Z}$, and no convenient system of generators is known.

7.2 Lipa’s Conjectures

Since the complex Hénon map is a diffeomorphism, it does not possess critical points in the usual sense. Therefore, one can not expect to obtain a “magic formula” as in item (ix) in Sect. 1 for the complex Hénon family. Here we propose two conjectures following the thesis of Lipa (2009) concerning the dynamics-parameter correspondence in the complex Hénon family. To do this, we investigate detailed combinatorial structure of the Mandelbrot set $\mathcal {M}$ based on Theorem 1.26.

Let H be a hyperbolic component of the Mandelbrot set $\mathcal {M}$. Theorem 1.26 implies that the union $R_{\mathcal {M}}(\theta _{H}^-)\cup R_{\mathcal {M}}(\theta _{H}^+)\cup \{r_{\mathcal {M}}(H)\}$ divide the complex plane into two parts when $H\ne \heartsuit$, and we also have $R_{\mathcal {M}}(\theta _{\heartsuit }^-)\cup R_{\mathcal {M}}(\theta _{\heartsuit }^+)\cup \{r_{\mathcal {M}}(\heartsuit )\}=[1/4, +\infty )$.

Definition 7.2

The connected component of $\mathbb {C}{\setminus } (R_{\mathcal {M}}(\theta _{H}^-)\cup R_{\mathcal {M}}(\theta _{H}^+)\cup \{r_{\mathcal {M}}(H)\})$ containing H is called the wake associated with H and denoted by $\mathcal {W}_{H}$.

One can associate the notion of a kneading sequence with each wake as follows Lau and Schleicher (1994), Lipa (2009) whose idea originates in Milnor–Thurston theory for maps of the interval (Milnor and Thurston 1977). Let H be a hyperbolic component and let k(H) be the period of the unique attractive cycle of $p_c$ with $c\in H$. Given $\theta \in \mathbb {T}$, we define $K_{H}^+(\theta )=(i^+_n)_{n\ge 0}\in \Sigma _2^{\mathbb {N}_0}$ as

$$\begin{aligned} i^+_n\equiv {\left\{ \begin{array}{ll} A &{} \text{ if } \, \, 2^{n}\theta \in [\frac{\theta +1}{2}, \frac{\theta }{2}) \\ B &{} \text{ if } \, \, 2^{n}\theta \in [\frac{\theta }{2}, \frac{\theta +1}{2}), \end{array}\right. } \end{aligned}$$

for $n\ge 0$, and $K_{H}^-(\theta )=(i^-_n)_{n\ge 0}\in \Sigma _2^{\mathbb {N}_0}$ as

$$\begin{aligned} i^-_n\equiv {\left\{ \begin{array}{ll} A &{} \text{ if } \, \, 2^n\theta \in (\frac{\theta +1}{2}, \frac{\theta }{2}] \\ B &{} \text{ if } \, \, 2^n\theta \in (\frac{\theta }{2}, \frac{\theta +1}{2}]. \end{array}\right. } \end{aligned}$$

for $n\ge 0$. One can then show that $K^+(\theta _{H}^-)=K^-(\theta _{H}^+)$ holds if $H\ne \heartsuit$.

Definition 7.3

The kneading sequence of the wake $\mathcal {W}_{H}$ associated with $H\ne \heartsuit$ is the first k(H) letters of the sequence $K^+(\theta ^-_{H})=K^-(\theta ^+_{H})$ and is denoted by $K(\mathcal {W}_{H})$, where k(H) is the period of the unique attractive cycle of $p_c$ for $c\in H$. We also set $K(\mathcal {W}_{\mathcal {\heartsuit }})\equiv A$.

We define the discarded kneading sequence of the wake $\mathcal {W}_{H}$ as the first $k(H)-1$ letters of the sequence $K^+(\theta ^-_{H})=K^-(\theta ^+_{H})$ and denote it by $\widehat{K}(\mathcal {W}_{H})$. We also set $\widehat{K}(\mathcal {W}_{\mathcal {\heartsuit }})$ to be the empty word $\epsilon$.

Here is a list of examples:

• When $H=\heartsuit$ is the Main Cardioid, we have $\theta ^-_{\heartsuit }=0$, $\theta ^+_{\heartsuit }=1$ and $k(\heartsuit )=1$. Since $K(\mathcal {W}_{\heartsuit })=A$ holds, we see $\widehat{K}(\mathcal {W}_{\heartsuit })=\epsilon$.

• When H is the Basilica component, we have $\theta ^-_{H}=1/3$, $\theta ^+_{H}=2/3$ and $k(H)=2$. Since $K(\mathcal {W}_{H})=BA$ holds, we see $\widehat{K}(\mathcal {W}_{H})=B$.

• When H is the Rabbit component, we have $\theta ^-_{H}=1/7$, $\theta ^+_{H}=2/7$ and $k(H)=3$. Since $K(\mathcal {W}_{H})=BBA$ holds, we see $\widehat{K}(\mathcal {W}_{H})=BB$.

• When H is the Airplane component, we have $\theta ^-_{H}=3/7$, $\theta ^+_{H}=4/7$ and $k(H)=3$. Since $K(\mathcal {W}_{H})=BAA$ holds, we see $\widehat{K}(\mathcal {W}_{H})=BA$.

Another consequence of Theorem 1.26 is that the Mandelbrot set $\mathcal {M}$ has a tree-like structure. More precisely, it yields that either $\mathcal {W}_{H}\supset \mathcal {W}_{H'}$, $\mathcal {W}_{H}\subset \mathcal {W}_{H'}$ or $\mathcal {W}_{H}\cap \mathcal {W}_{H'}=\emptyset$ holds for two hyperbolic components H and $H'$ of $\mathcal {M}$.

Definition 7.4

Let H and $H'$ be two hyperbolic components of $\mathcal {M}$. We say that the wake $\mathcal {W}_{H'}$ is conspicuous to the wake $\mathcal {W}_{H}$ if

• (1) $\mathcal {W}_{H'}\subset \mathcal {W}_{H}$,

• (2) $k(H')\le k(H)$,

• (3) there are no hyperbolic components $H''$ with $k(H'')<k(H')$ and $\mathcal {W}_{H'}\subset \mathcal {W}_{H''}\subset \mathcal {W}_{H}$.

We remark that a wake is always conspicuous to itself.

Through her numerical experiments with SaddleDrop, S. Koch (Lipa 2009; Koch 2012) observed the “splitting phenomenon” of the Mandelbrot set $\mathcal {M}$. Based on this phenomenon she defined (naively) the notion of herds as follows.

Suppose first that $b=0$ and look at the c-plane in the parameter space (see the left picture in Fig. 17); we then see the Mandelbrot set $\mathcal {M}$ in the c-plane. If we change b slightly, we still have a Mandelbrot-like set in the corresponding c-plane to which we still have a well-defined notion of wakes.

Let $\mathcal {W}_{H}$ be a wake of the Mandelbrot-like set with $k(H)= 2$. As we perturb b more, the wake $\mathcal {W}_{H}$ seems to split into two different pieces (see the middle picture in Fig. 17). One, called the A-herd of $\mathcal {W}_{H}$, which contains all the wakes in $\mathcal {W}_{H}$ whose discarded kneading sequences end in A, moves to the direction in the c-plane that b is perturbed in. The other, called the B-herd of $\mathcal {W}_{H}$, which contains all the wakes in $\mathcal {W}_{H}$ whose discarded kneading sequence end in B, moves in the opposite direction (see the right picture in Fig. 17).

Fig. 17

The c-planes with $b=0$ (left), $b=0.015i$ (middle) and $b=0.05i$ (right) (Lipa 2009)

Fig. 18

A loop surrounding the B-herd of Airplane wake with $b=0.05i$ ((left) which splits to the BB-herd and the AB-herd with $b=0.2+0.3i$ (middle) (Lipa 2009)

Let $\mathcal {W}_{H'}$ be a wake of the Mandelbrot-like set with $k(H')= 3$ and $\mathcal {W}_{H'}\subset \mathcal {W}_{H}$. As we perturb b even more, the A-herd of $\mathcal {W}_{H}$ inside $\mathcal {W}_{H'}$ splits into two pieces. One, called the AA-herd of $\mathcal {W}_{H'}$, which contains all the wakes in $\mathcal {W}_{H'}$ whose discarded kneading sequences end in AA, moves a bit farther to the direction in the c-plane that b is perturbed in than the other, called the BA-herd of $\mathcal {W}_{H'}$, which contains all the wakes in $\mathcal {W}_{H'}$ whose discarded kneading sequence end in BA. Similarly, the B-herd of $\mathcal {W}_{H}$ inside $\mathcal {W}_{H'}$ (see the left picture in Fig. 18) splits into two pieces; the AB-herd of $\mathcal {W}_{H'}$ and the BB-herd of $\mathcal {W}_{H'}$ (see the right picture in Fig. 18).

After one more splitting, we have 8 herds associated with discarded kneading sequences in the following order: AAB, BAB, BBB, ABB, ABA, BBA, BAA, AAA (note that the parity of the number of the letter B flips the lexicographical order). In this way we obtain the notion of the $\underline{v}$-herd of a wake $\mathcal {W}_{H}$ for a word $\underline{v}$ over the alphabet $\Sigma _2=\{A, B\}$. According to Lipa (2009), this splitting phenomenon has been observed numerically using SaddleDrop to a depth of 5.

Below the length of a word $\underline{w}$ over the alphabet $\{A, B, *\}$ is denoted by $|\underline{w}|$. Given a word $\underline{w}$ over $\{A, B, *\}$ containing exactly one $*$, we will define a continuous map $\tau _{\underline{w}} : \Sigma _2^{\mathbb {Z}} \rightarrow \Sigma _2^{\mathbb {Z}}$ as follows. Take a sequence $\underline{\varepsilon }=(\varepsilon _n)_{n\in \mathbb {Z}}\in \Sigma _2^{\mathbb {Z}}$. If there exist $k\in \mathbb {Z}$ with $\varepsilon _k\cdots \varepsilon _{k+|\underline{w}|-1}=\underline{w}$ except for the digit of $*$ in $\underline{w}$, we replace the letter in the corresponding digit in $\underline{w}$ to the opposite one (i.e. A to B and B to A). If there is no such $k\in \mathbb {Z}$, $\underline{\varepsilon }$ is left unchanged. By operating the above procedure to all possible $k\in \mathbb {Z}$, we obtain a new sequence denoted by $\tau _{\underline{w}}(\underline{\varepsilon })\in \Sigma _2^{\mathbb {Z}}$.

Definition 7.5

Let $W\equiv \{\underline{w}^1, \dots , \underline{w}^m\}$ be a set of finite words over $\{A, B, *\}$, each containing exactly one $*$. Assume that $\tau _{\underline{w}^1}, \dots , \tau _{\underline{w}^m}$ are all commutative with each other with respect to the composition of the maps. Then,

$$\begin{aligned} \tau _W \equiv \tau _{\underline{w}^m}\circ \cdots \circ \tau _{\underline{w}^1} : \Sigma ^{\mathbb {Z}}_2 \longrightarrow \Sigma ^{\mathbb {Z}}_2 \end{aligned}$$

is called a compound marker endomorphism given by W. If a compound marker endomorphism is an automorphism, it is called a compound marker automorphism.

Now we are in position to state the first conjectures of Lipa (Conjecture 8.3 in Lipa 2009).

Conjecture 7.6

Let $\mathcal {W}_{H}$ be a wake with its conspicuous wakes $\mathcal {W}_{H_1}, \dots , \mathcal {W}_{H_m}$ and let $\underline{v}$ be a word over $\{A, B\}$. Suppose that $\gamma \in \pi _1(\mathcal {S}_{c, b}, (c_0, b_0))$ winds around the $\underline{v}$-herd of the wake $\mathcal {W}_{H}$ and let $\underline{w}^i\equiv \underline{v}*K(\mathcal {W}_{H_i})$. Then, $\tau _{\underline{w}^1}, \dots , \tau _{\underline{w}^m}$ are all commutative and the compound marker endomorphism $\tau _W$ given by $W\equiv \{\underline{w}^1, \dots , \underline{w}^m\}$ coincides with $\rho (\gamma )$. In particular, it is an automorphism.

When $\mathcal {W}_{H}$ is the Airplane wake, we have $K(\mathcal {W}_{H})=BAA$. He observed that the monodromy action $\rho (\gamma )$ along the path $\gamma$ in the left picture of Fig. 18 coincides with $\tau _W$, where $W=\{B*BAA\}$ (see Subsection 9.1 of Lipa 2009). Similarly, he observed such coincidence of $\rho (\gamma )$ for the two loops in the right picture of Fig. 18 and $\tau _W$ with $W=\{BB*BAA\}$ and $W=\{AB*BAA\}$ respectively (see Subsection 9.2 of Lipa 2009).

In Conjecture 8.4 of Lipa 2009, Lipa proposed the following “converse” to Conjecture 7.6.

Conjecture 7.7

Let $\mathcal {W}_{H}$ be a wake with its conspicuous wakes $\mathcal {W}_{H_1}, \dots , \mathcal {W}_{H_m}$ and $\underline{v}$ be a word over $\{A, B\}$. Let $\underline{w}^i\equiv \underline{v}*K(\mathcal {W}_{H_i})$ and assume that $\tau _{\underline{w}^1}, \dots , \tau _{\underline{w}^m}$ are all commutative. Suppose that the compound marker endomorphism $\tau _W$ given by $W\equiv \{\underline{w}^1, \dots , \underline{w}^m\}$ is not an automorphism. Then, there is no $\gamma \in \pi _1(\mathcal {S}_{c, b}, (c_0, b_0))$ which winds around the $\underline{v}$-herd of $\mathcal {W}_{H}$.

For example, Lipa showed that $\tau _W$ with $W=\{A*BAA\}$ is not a compound marker automorphism and claims that he was not able to find a loop in the horseshoe locus that winds only around the A-herd of the Airplane wake in Subsection 9.3 of Lipa (2009). See Subsections 9.5 and 9.6 of Lipa (2009) for more examples and related discussions.

7.3 Visualization in $\mathbb {C}^2$

There are several programs to draw the Julia sets and parameter loci of the Henon family $f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ in $\mathbb {C}^2$ which are extremely useful for theoretical considerations of the dynamics. FractalAsm (2000) and SaddleDrop (2000) are created by Cornell Dynamics group (see Sect. 3.1). Ushiki (2012) made programs called HénonExplorer and StereoViewer; Figs. 1, 19, 20, 21 and 22 are drawn by these programs. These pictures present sets in four-dimensional space, and this software helps to visualize these pictures by showing them under different projections to the two-dimensional computer screen, or onto a stereo 3D viewer. An important feature is that the viewer allows the set to be viewed as it is rotated in 4D space. We recommend to visit his webpage (Ushiki 2012) where more images can be found.

Figures 1 and 19 are drawn by HénonExplorer and Figs. 20, 21 and 22 are drawn by StereoViewer. Each of these figures shows a “cloud” of points which are colored whitish blue. These are the periodic points of periods up to 20 for a Hénon map f. Asymptotically, most periodic points are of saddle type, and the asymptotic distribution of the saddle periodic points gives the unique maximal entropy measure $\mu _f$ of f (see Bedford et al. 1993b). Thus this cloud of points approximates the set $J_f^{*}$, which is defined as the support of the measure $\mu _f$ (see Footnote 5 in Sect. 2.2). For the maps of Figs. 1 and 19, there are two saddle fixed points, and the surfaces shown in the figures are portions of the stable and unstable manifolds. Note that if we wish to display stable or unstable manifolds, we need to truncate them. This is because the stable and unstable manifolds of any saddle periodic point have homoclinic intersection, hence they return arbitrary close to the periodic point (see Bedford et al. 1993a).

Fig. 19

Close to a heteroclinic tangency (Ushiki 2012)

In Fig. 1 there are two saddle fixed points close to each other which are bifurcated from a parabolic fixed point. The rectangular surface colored orange/brown is a piece of the stable manifold of one of the two saddle fixed points (there is a similar cyan-colored region, corresponding to the other saddle fixed point.) The darker shading corresponds to the values of $G^-$; the darkest part indicates points of $J^-$, where $G^-(x, y)=0$. The other two surfaces are portions of the unstable manifolds, cut off so as not to obscure other portions of the picture. The shading corresponds to the value of $G^+$, with the darkest part showing the points of $J^+$. These saddle points in Fig. 1 are bifurcated from a map with a semi-parabolic fixed point. This bifurcation corresponds to a “parabolic implosion”, which we can see because the fixed points show the spiraling behavior seen with the parabolic explosion of the familiar one-dimensional “cauliflower” Julia set. Such bifurcations were studied in detail by Bedford et al. (2012), and it is interesting to compare Fig. 1 with the figures in Bedford et al. (2012).

Figure 19 describes the dynamics of a Hénon map which has an almost tangential heteroclinic intersection. In the figure, the truncated surface colored green/yellow/red surrounding the one-dimensional-like filled Julia set in purple represents a part of the unstable manifold of a saddle fixed point. The other surface colored pink, which looks like an arch, represents the stable manifold of another saddle fixed point. As in Fig. 1, the darkness of the shading corresponds to the value of $G^{\pm }$, and the darkest part indicates points of $J^{\pm }$ where the surfaces intersect with the whitish blue cloud of points. The two surfaces intersect with each other almost tangentially at the “saddle point” of the arch, but this intersection does not belong to $\mathbb {R}^2$.

Fig. 20

Hénon map at the classical parameter (Ushiki 2012)

Figure 20 describes the support $J_f^{*}$ of the maximal entropy measure for the Hénon map $f_{a, b}$ at the classical parameter $(a, b)=(1.4, -0.3)$ (Hénon 1976) seen from different directions. In each figure one can find the well-known strange attractor embedded in the picture which is the closure of the real unstable manifold of a saddle fixed point in the first quadrant of $\mathbb {R}^2$. The attractor is decorated with portions of $J_f^{*}$ not in the real plane. These are the “pruned branches” emanating into the imaginary directions. When we change the direction of our view-point as in the figures, these directions are twisted unexpectedly. These pruned branches become smaller when the parameter a increases, and eventually disappear when $f_{a, b}$ becomes a horseshoe on $\mathbb {R}^2$.

Figure 21 describes the dynamics of a real Hénon map which preserves the area in $\mathbb {R}^2$. The large red curves are the KAM invariant circles of period two sitting in the real plane on which the twice iterate of $f_{a, b}$ is topologically conjugate to an irrational rotation. One can also observe the nested structure of smaller KAM circles in red and the so-called chaotic sea in purple/yellow where the Lyapunov exponent is conjecturally strictly positive. Blue and green points represent orbits of randomly chosen initial points near the complex extensions of the KAM circles. They seem to present quasi-periodic motions, but we do not know if their orbit closures form tori.

Fig. 21

Volume preserving Hénon map with KAM circles (Ushiki 2012)

Fig. 22

Hénon map with coexisting attractive cycles (Ushiki 2012)

Figure 22 describes $J_f^{*}$ of a Hénon map which possesses two attractive cycles of periods one and three respectively. The set $J_f^{*}$ looks disconnected and hyperbolic (since it seems to satisfy the cigar condition; see Bedford and Smillie 1999 for more details). The largest circle in $J_f^{*}$ as well as its images belong to the boundary of the attractive basin of period one, and the other circles belong to the boundary of the attractive basins of period three. Locally $J_f^{*}$ looks like the product of a portion of a one-dimensional-like Julia set and a Cantor set. However, since this map is quadratic and has two attractive cycles, one can show that it is non-planar (by assuming its hyperbolicity).

Let us explain some mathematical background behind Ushiki’s programs. In these programs we first compute periodic points of the Hénon map f of period m, say $m\le 20$. This means that we need to find all points $(x, y)\in \mathbb {C}^2$ satisfying $f^m(x, y)=(x, y)$. However, this is a polynomial equation of degree $2^m$ and, when m is large, it is practically impossible to find the zeros of the equation with such large degree by Newton’s root-finding algorithm because the size of the attractive basins for the solutions can be extremely small.

To overcome this difficulty, Ushiki employed an algorithm of Biham and Wenzel (1989, 1990), now called the BW-algorithm. For a given symbol sequence $(\varepsilon _n)_{n\in \mathbb {Z}}\in \{+1, -1\}^{\mathbb {Z}}$, the algorithm consists of the following infinitely many ordinary differential equations:

$$\begin{aligned} \frac{d}{dt}s_n(t)=\varepsilon _n \cdot \{s_{n+1}(t)-s_n(t)^2+a+bs_{n-1}(t)\}, \end{aligned}$$

(4)

where $s_n(t)$ is a $\mathbb {C}$-valued function. Note that an equilibrium $(s^{*}_n)_{n\in \mathbb {Z}}$ of the system (4) satisfies the recursion equation $s^{*}_{n+1}=(s^{*}_n)^2-a-bs^{*}_{n-1}$, which is equivalent to $(s^{*}_{n+1}, s^{*}_n)=f^n_{a, b}(s^{*}_1, s^{*}_0)$. In particular, by taking $(\varepsilon _n)_{n\in \mathbb {Z}}$ as a periodic sequence, an equilibrium of (4) corresponds to a periodic point of $f_{a, b}$. Biham and Wenzel (1989, 1990) conjectured that (4) has a unique (globally attracting) equilibrium $(s^{*}_n)_{n\in \mathbb {Z}}$ for any $(\varepsilon _n)_{n\in \mathbb {Z}}\in \{+1, -1\}^{\mathbb {Z}}$ and $(a, b)\in \mathbb {C}\times \mathbb {C}^{\times }$, and the correspondence $(\varepsilon _n)_{n\in \mathbb {Z}}\mapsto (s^{*}_n)_{n\in \mathbb {Z}}$ from $\{+1, -1\}^{\mathbb {Z}}$ to the space of all bounded bi-infinite orbits of $f_{a, b}$ in $\mathbb {C}^2$ is bijective. Although a counter-example was found for $(a, b)=(1, 0.54)$ in Grassberger et al. (1989) where (4) has a limit cycle for certain $(\varepsilon _n)_{n\in \mathbb {Z}}$, the BW-algorithm is practically a very useful algorithm to find periodic points. It would be interesting to give a rigorous proof of its convergence for reasonable initial data $(s_n(0))_{n\in \mathbb {Z}}$ and appropriate choice of parameter (a, b). Note that Sterling and Meiss (1998) justified the convergence of the algorithm in $\mathbb {R}^2$ for $a>0$ large by employing the idea of anti-integrable limits and Mummert (2008) extended their convergence result to other cases in $\mathbb {C}^2$.

As we saw above, the images drawn by HénonExplorer/StereoViewer are presented as objects in the two/three-dimensional space. However, the actual Julia sets are in $\mathbb {C}^2$ which has real dimension four. As we observe especially through Fig. 20, it is hard to imagine how they are sitting in the four-dimensional space. Recently we have launched a 4D visualization project called Watch $\_$ H ¹⁰. The goal of the project is to express the images related to complex dynamics as fractal objects in the four-dimensional space and make an archive of such images (dynamical and parameter spaces for the Hénon family, dynamics on complex surfaces, etc). Towards this goal, we plan to employ a 3D virtual reality system, analyze the rotation in the four-dimensional space and develop new rendering techniques adapted to it.

8 Applications to the Dynamics of Hénon Maps in $\mathbb {R}^2$

In this section we discuss the problems (x). In Sect. 8.1 we study global topology of two real parameter loci and in Sect. 8.2 we investigate local geometry of their boundaries and apply it to the study of ground states at “temperature zero”.

8.1 Two Real Loci

Let f be a polynomial diffeomorphism of $\mathbb {C}^2$ and let $d\ge 2$ be its degree. We say that f is real if all the coefficients of f are real. In this case, the restriction $f|_{\mathbb {R}^2} : \mathbb {R}^2\rightarrow \mathbb {R}^2$ is a well-defined dynamical system. It is known (Friedland and Milnor 1989) that the topological entropy of $f|_{\mathbb {R}^2}$ satisfies $0\le h_{\mathrm {top}}(f|_{\mathbb {R}^2})\le \log d$ for any real f of degree d. We therefore say that a real polynomial diffeomorphism f attains the maximal entropy if $h_{\mathrm {top}}(f|_{\mathbb {R}^2})=\log d$.

In Bedford and Smillie (2002) developed the theory of quasi-hyperbolicity. An important example of a quasi-hyperbolic map is a real polynomial diffeomorphism f with maximal entropy. Based on this theory, they have solved the so-called “first tangency problem” as follows.

Theorem 8.1

(Bedford and Smillie 2004) Assume that a real polynomial diffeomorphism f attains the maximal entropy. Then, either $f|_{\mathbb {R}^2}$ is uniformly hyperbolic on the non-wandering set $\Omega (f|_{\mathbb {R}^2})$ or has a tangency between stable and unstable manifolds.

Hereafter, we restrict our attention to the following form of the Hénon family:

$$\begin{aligned} f_{a, b} : (x, y)\longmapsto (x^2-a-by, x), \quad (a, b)\in \mathbb {R}\times \mathbb {R}^{\times } \end{aligned}$$

as dynamical systems on $\mathbb {R}^2$. Let us call $\mathbb {R}\times \mathbb {R}^{\times }$ the parameter space for the real Hénon family $f_{a, b}$. When $b\ne 0$ is fixed and a is large enough, $f_{a, b}$ is a hyperbolic horseshoe on $\mathbb {R}^2$, i.e. the restriction of $f_{a, b}$ to its non-wandering set is uniformly hyperbolic and is topologically conjugate to the full shift with two symbols (Devaney and Nitecki 1979). Such $f_{a, b}$ attains the maximal entropy among the Hénon maps since we know $0\le h_{\mathrm {top}}(f_{a, b})\le \log 2$ for any $(a, b)\in \mathbb {R}\times \mathbb {R}^{\times }$ by Friedland and Milnor (1989).

We are thus led to introduce the hyperbolic horseshoe locus:

$$\begin{aligned} \mathcal {H}_{\mathbb {R}}\equiv \bigl \{(a, b) \in \mathbb {R}\times \mathbb {R}^{\times } : f_{a, b} \text{ is } \text{ a } \text{ hyperbolic } \text{ horseshoe } \text{ on } \mathbb {R}^2 \bigr \} \end{aligned}$$

as well as the maximal entropy locus:

$$\begin{aligned} \mathcal {M}_{\mathbb {R}}\equiv \bigl \{(a, b) \in \mathbb {R}\times \mathbb {R}^{\times } : f_{a, b} \text{ attains } \text{ the } \text{ maximal } \text{ entropy }\log 2 \bigr \}. \end{aligned}$$

Note that $\mathcal {H}_{\mathbb {R}}$ is open and $\mathcal {M}_{\mathbb {R}}$ is closed in $\mathbb {R}\times \mathbb {R}^{\times }$ (since $h_{\mathrm {top}}(f_{a, b})$ is a continuous function of (a, b) by results of Katok and Newhouse; see page 110 of Milnor 1988), hence $\overline{\mathcal {H}_{\mathbb {R}}}\subset \mathcal {M}_{\mathbb {R}}$.

Theorem 8.2

(Bedford and Smillie 2006; Arai and Ishii 2015) There exists a real analytic function $a_{\mathrm {tgc}} : \mathbb {R}^{\times } \rightarrow \mathbb {R}$ from the b-axis to the a-axis of the parameter space for the Hénon family $f_{a, b}$ with $\lim _{b\rightarrow 0}a_{\mathrm {tgc}}(b)=2$ so that

• (i) $(a, b)\in \mathcal {H}_{\mathbb {R}}$ iff $a>a_{\mathrm {tgc}}(b)$,

• (ii) $(a, b)\in \mathcal {M}_{\mathbb {R}}$ iff $a\ge a_{\mathrm {tgc}}(b)$.

This result has been first obtained by Bedford and Smillie (2006) for the case $|b|<0.06$ and then generalized to all $b\ne 0$ by Arai and the author (Arai and Ishii 2015). We note that, when $a=a_{\mathrm {tgc}}(b)$, the map $f_{a, b}$ has exactly one orbit of either homoclinic ($b>0$) or heteroclinic ($b<0$) tangencies of stable and unstable manifolds of suitable saddle fixed points (Bedford and Smillie 2004). The strategy of Bedford and Smillie (2006), Arai and Ishii (2015) is first to extend the dynamical and the parameter spaces over $\mathbb {C}$, investigate their complex dynamical and complex analytic properties, and then reduce them to obtain conclusions over $\mathbb {R}$. In the article (Arai and Ishii 2015) we also employ interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences.

The statements described in Theorem 8.2 justify what were numerically observed at the beginning of 1980’s by El Hamouly and Mira, Tresser, Ushiki and others. Figure 23 is obtained by joining two figures in the numerical work of El Hamouly and Mira (1981) and turning it upside down. There, the graph of the function $a_{\mathrm {tgc}}$ is implicitly figured out by the right-most wedge-shaped curve.

As a consequence of Theorem 8.2, we obtain some global topological properties of the two loci. To state them, let us put $\mathcal {H}^{\pm }_{\mathbb {R}}\equiv \mathcal {H}_{\mathbb {R}}\cap \{\pm b>0\}$ and $\mathcal {M}^{\pm }_{\mathbb {R}}\equiv \mathcal {M}_{\mathbb {R}}\cap \{\pm b>0\}$. Below, we take the closure and the boundary of $\mathcal {H}^{\pm }_{\mathbb {R}}$ and $\mathcal {M}^{\pm }_{\mathbb {R}}$ in $\{\pm b>0\}$.

Corollary 8.3

(Arai and Ishii 2015) Both loci $\mathcal {H}^{\pm }_{\mathbb {R}}$ and $\mathcal {M}^{\pm }_{\mathbb {R}}$ are connected and simply connected in $\{\pm b>0\}$. Moreover, we have $\overline{\mathcal {H}^{\pm }_{\mathbb {R}}}=\mathcal {M}^{\pm }_{\mathbb {R}}$ and $\partial \mathcal {H}^{\pm }_{\mathbb {R}}=\partial \mathcal {M}^{\pm }_{\mathbb {R}}$.

Fig. 23

Bifurcation curves of the Hénon family El Hamouly and Mira (1981)

We note that this corollary can be regarded as a first step towards the understanding of an “ordered structure” in the parameter space for the Hénon family. Recall that in Milnor and Tresser (200) the monotonicity of the topological entropy for the cubic family (which has two parameters) is formulated as the connectivity of the isentropes. Therefore, the above result indicates a weak form of monotonicity of the entropy function $(a, b)\mapsto h_{\mathrm {top}}(f_{a, b})$ at its maximal value.

It should be interesting to compare our results to the so-called anti-monotonicity theorem in Kan et al. (1992). To be precise, we let $h_t : \mathbb {R}^2\rightarrow \mathbb {R}^2$ ($t\in \mathbb {R}$) be a one-parameter family of dissipative $C^3$-diffeomorphisms of the plane and assume that $h_{t_0}$ has a non-degenerate homoclinic tangency at certain parameter $t=t_0$. Then, there are both infinitely many orbit-creation parameters and infinitely many orbit-annihilation parameters in any neighborhood of $t_0\in \mathbb {R}$. It has been shown in Bedford and Smillie (2006) that for the one-parameter family of Hénon maps $\{f_{a, b_{*}}\}_{a\in \mathbb {R}}$ with a fixed $b_{*}>0$ sufficiently close to zero, the map at $a=a_{\mathrm {tgc}}(b_{*})$ taken from the boundary $\partial \mathcal {H}^+_{\mathbb {R}}=\partial \mathcal {M}^+_{\mathbb {R}}$ has a non-degenerate homoclinic tangency. Of course, anti-monotonicity of some orbits does not necessarily imply anti-monotonicity of topological entropy. Nonetheless, the anti-monotonicity theorem suggests that, a priori, both $\mathcal {H}_{\mathbb {R}}$ and $\mathcal {M}_{\mathbb {R}}$ might have holes or several connected components separated from the largest one described in Corollary 8.3.

In their recent work Bedford and Smillie 2017 gave a characterization of the loci boundary for $|b|<0.06$ in terms of symbolic dynamics with respect to a family of three polydisks. A similar characterization of the loci boundary should be possible for all values of b in terms of symbolic dynamics with respect to a family of four polydisks for $b>0$ and a family of five polydisks for $b<0$ constructed in Arai and Ishii (2015).

Further problems and questions follow:

• Is the function $a_{\mathrm {tgc}}$ monotone on $\{b>0\}$ and on $\{b<0\}$?

• Is the boundary of the zero-entropy locus piecewise real analytic (see Gambaudo et al. 1991)? This is true when |b| is close to zero (Gambaudo et al. 1989; Carvalho et al. 2005).

• Is the complex hyperbolic horseshoe locus $\mathcal {H}_{\mathbb {C}}$ for the complex Hénon family connected? We already know that it is not simply connected (see Sect. 7.1). Remark that since $h_{\mathrm {top}}(f_{a, b})=\log 2$ for any $(a, b)\in \mathbb {C}\times \mathbb {C}^{\times }$ by Theorem 2.2, the complex maximal entropy locus $\mathcal {M}_{\mathbb {C}}$ is the entire parameter space $\mathbb {C}\times \mathbb {C}^{\times }$.

8.2 Ground States

One of the key steps in the proof of Theorem 8.2 was to show that the boundaries of the first tangency locus $\partial \mathcal {H}^{\pm }_{\mathbb {R}}$ is surrounded by “tin cans” in the complexified parameter space (see Subsection 5.2 in Arai and Ishii 2015). This condition has been verified by employing the interval Krawczyk method, an interval arithmetic version of Newton’s root-finding algorithm. By modifying this argument together with the Schwarz Lemma in the parameter space we obtain the following estimate on the derivative of the function $a_{\mathrm {tgc}}$.

Theorem 8.4

(Arai et al. 2017) We have

$$\begin{aligned} \frac{9}{8}<\lim _{b\rightarrow +0}\frac{da_{\mathrm {tgc}}}{db}(b)<\frac{23}{8}. \end{aligned}$$

Theorem 8.4 is applied to investigate ergodic properties of the real Hénon maps $f_{a, b}$ at the first bifurcation parameters $(a, b)\in \partial \mathcal {H}_{\mathbb {R}}^+$. Among others, we are interested in a variational characterization of equilibrium measures “at temperature zero”. To state it, denote by M(f) the space of f-invariant Borel probability measures of a Hénon map f. An invariant measure $\mu \in M(f)$ is called a $(+)$-ground state if there exists an increasing sequence $t_n\in \mathbb {R}$ with $t_n\rightarrow +\infty$ as $n\rightarrow \infty$ so that $\mu$ is obtained as the weak limit of equilibrium measures for the potential function $-t_n\log \Vert D_pf|E^u_p\Vert$, where $E^u_p$ is the unstable direction of $D_pf$ at $p\in \mathbb {R}^2$. Let

$$\begin{aligned} \Lambda ^u_{\mu }(f)\equiv \int \log \Vert D_zf|E^u_p\Vert d\mu (p), \end{aligned}$$

be the unstable Lyapunov exponent of f with respect to $\mu \in M(f)$ and let

$$\begin{aligned} \Lambda ^u(a, b)\equiv \inf _{\nu \in M(f_{a, b})}\Lambda ^u_{\nu }(f_{a, b}). \end{aligned}$$

An invariant measure $\mu \in M(f)$ is called Lyapunov minimizing if it attains the infimum above. An invariant measure $\mu \in M(f)$ is called entropy maximizing among the Lyapunov minimizing measures if it attains the supremum of the metric entropy $h_{\nu }(f)$ over all Lyapunov minimizing measures $\nu$ (see Takahasi 2016 for more detail).

Let $U_{\delta }$ be the $\delta$-neighborhood of the Chebyshev point $(a, b)=(2, 0)$ in the parameter space of the real Hénon family $f_{a, b}$. One can see that Theorem 8.4 yields a non-degeneracy condition in Theorem A (a) of Takahasi (2016) for the Hénon maps $f_{a, b}$ with $(a, b)\in \partial \mathcal {H}^+_{\mathbb {R}}\cap U_{\delta }$. As a consequence, we have the following variational characterization of the $(+)$-ground states.

Corollary 8.5

(Arai et al. 2017) There exists $\delta >0$ so that any $(+)$-ground state of any Hénon map $f_{a, b}$ with $(a, b)\in \partial \mathcal {H}^+_{\mathbb {R}}\cap U_{\delta }$ is Lyapunov minimizing, and entropy maximizing among the Lyapunov minimizing measures.

Corollary 8.5 indicates that a local geometric property of a complex parameter locus yields ergodic property of real Hénon maps.