Lyapunov Exponents and Nonadapted Measures for Dispersing Billiards

Abstract

For hyperbolic systems with singularities, such as dispersing billiards, Pesin theory as developed by Katok and Strelcyn applies to measures that are “adapted” in the sense that they do not give too much weight to neighborhoods of the singularity set. The zero-entropy measures supported on grazing periodic orbits are nonadapted, but it has been an open question whether there are nonadapted measures with positive entropy. We construct such measures for any dispersing billiard with a periodic orbit having a single grazing collision; we then use our construction to show that the thermodynamic formalism for such billiards has a phase transition even when one restricts attention to adapted or to positive entropy measures.

1 Introduction

Sinai introduced dispersing billiards as a mathematically tractable example of a mechanical system where the Boltzmann hypothesis can be verified [S]. Since then, mathematical billiards, and the Lorentz gas in particular, have become central models in mathematical physics. Although the billiard map associated with a Lorentz gas has discontinuities, it preserves a smooth invariant measure, which we denote by $\mu _{ \text{ SRB }}$. Classical results focus on establishing quantitative statistical properties such as the Central Limit Theorem, decay of correlations and large deviation estimates with respect to $\mu _{ \text{ SRB }}$, and they use a variety of techniques, including Markov partitions and sieves [BSC1, BSC2], Young towers [Y], spectral analysis of the transfer operator [DZ] and, most recently, projective cones [DL].

In recent years, there has been significant interest in the existence and properties of other invariant measures for the billiard map. In particular, the existence and uniqueness of equilibrium measures with respect to the geometric family of potentials, $- t \log J^uT$, $t \ge 0$, where $J^uT$ denotes the unstable Jacobian of the billiard map T, are studied in [BD1] (for $t=0$) and [CWZ, BD2] (for $t>0$). A renewed interest in the construction of Markov partitions for surface diffeomorphisms stemming from [Sar] has also led to symbolic codings for more general hyperbolic measures for billiards [LM, ALP]. These constructions in turn lead to more questions regarding properties of these invariant measures, such as possible Lyapunov spectra and phase transitions in the associated pressure functions.

For a class of planar dispersing billiard maps corresponding to a finite horizon Lorentz gas, we present a construction that answers two open questions in the billiards literature. The first relates to the construction of countable Markov partitions for billiards, which relies on Pesin theory as developed by Katok and Strelcyn; this theory applies to invariant probability measures $\mu$ that are adapted, meaning that

$$\begin{aligned} \int |\log d(x,{\mathcal {S}})| \,d\mu < \infty , \end{aligned}$$

(1.1)

where ${\mathcal {S}}$ is the singular set for T.¹ Using this theory, Lima and Matheus showed that if $\mu$ is adapted and hyperbolic, then there is a countable Markov partition for the billiard map T which provides a symbolic coding for $\mu$-almost every point [LM]. Since measures with large entropy are likely to be hyperbolic (this occurs e.g. when the Ruelle inequality holds), a natural question arises: are there T-invariant nonadapted measures with positive entropy? We answer this question in the affirmative.

Theorem 1.1

There are finite horizon planar dispersing billiards with invariant probability measures of positive entropy that are not adapted.

The second question is related to the behavior of the pressure function P(t) for the family of geometric potentials $- t \log J^uT$ for t near 0, where $J^uT$ denotes the unstable Jacobian of T. The pressure function is defined as

$$\begin{aligned} P(t) = \sup _{\mu \in {\mathcal {M}}} \left\{ h_\mu (T) - t \int \log J^uT \, d\mu \right\} , \end{aligned}$$

(1.2)

where ${\mathcal {M}}$ is the set of T–invariant probability measures. The function P(t) is finite and decreasing for $t \ge 0$, and [BD2] proved it is analytic on an interval $(0, t_*)$, for some $t_*>1$ depending on the billiard table. While $P(0) < \infty$ is the topological entropy of the map² [BD1], it is clear that for a table with a grazing periodic orbit, $P(t) = \infty$ for $t<0$ due to the invariant measure supported on such an orbit. We arrive at the following natural question: does one still see a jump in P(t) at $t=0$ if one restricts the supremum in (1.2) to a smaller class of measures? For example, letting $N_\varepsilon ({\mathcal {S}})$ denote the $\varepsilon$–neighborhood of ${\mathcal {S}}$, we can restrict the supremum to measures satisfying:

• (a) There are constants $C, \alpha >0$ such that $\mu (N_\varepsilon ({\mathcal {S}})) \le C\varepsilon ^\alpha$ for all $\varepsilon >0$ (such a measure must be adapted); or

• (b) $\mu$ has positive entropy.

Our construction answers this question in the affirmative for both restricted suprema under condition (a) or (b) whenever a certain type of periodic orbit exists on the billiard table. This implies that, for some classes of dispersing billiards, there is a phase transition at $t=0$.

Theorem 1.2

There are finite horizon planar dispersing billiards such that each of them possesses a sequence of periodic orbits $(p_n)$ with positive and finite Lyapunov exponents, but tending to infinity as $n \rightarrow \infty$. As a consequence, for all $t<0$ we have

$$\begin{aligned}{} & {} \sup _{\mu \in {\mathcal {M}}} \left\{ h_\mu (T) - t \int \log J^uT \, d\mu : \exists C, \alpha> 0 \text{ such } \text{ that } \mu (N_\varepsilon ({\mathcal {S}})) \le C\varepsilon ^\alpha ,\forall \varepsilon > 0 \right\} \nonumber \\{} & {} \quad = \infty . \end{aligned}$$

(1.3)

Moreover, it follows from Theorem 1.1 that for such dispersing billiards and for all $t < 0$ we have

$$\begin{aligned} \sup _{\mu \in {\mathcal {M}}} \left\{ h_\mu (T) - t \int \log J^uT \, d\mu : \mu \text{ has } \text{ positive } \text{ entropy } \right\} = \infty . \end{aligned}$$

(1.4)

The case of the infinite horizon Lorentz gas has been studied in [CT]. There, the authors show that $P(0) = \infty$ and construct invariant measures with both infinite metric entropy as well as invariant measures with finite entropy, but infinite Lyapunov exponent.

Remark 1.3

We remark that for $t>0$, restricting to measures that satisfy both (a) and (b) does not change the pressure P(t); specifically, the equilibrium states produced in [BD2] satisfy both conditions for all $t \in (0, t_*)$. However, it is not known whether the measure of maximal entropy $\mu _0$ (in the case $t=0$) satisfies (a). Indeed, [BD1] produces the weaker bound $\mu _0(N_\varepsilon ({\mathcal {S}})) \le C |\log \varepsilon |^{-\gamma }$ for some $\gamma >1$; this is sufficient for $\mu _0$ to be adapted and hyperbolic.

We state Theorem 1.2 using the stronger condition (a) since it makes for a stronger statement: even restricting to this smaller class of measures, the pressure is infinite when $t<0$.

2 Setting

Our billiard table is defined by placing finitely many pairwise disjoint closed, convex sets $O_i$, $i = 1, \ldots , d$, in ${\mathbb {T}}^2 = {\mathbb {R}}^2 / {\mathbb {Z}}^2$. We assume that the boundaries $\partial O_i$ are $C^3$ curves with strictly positive curvature. The billiard table is then defined as ${\mathcal {Q}}:= {\mathbb {T}}^2 {\setminus } (\bigcup _{i=1}^d O_i)$. The billiard flow is induced by the motion of a point particle traveling at unit speed in ${\mathcal {Q}}$ and reflecting elastically at collisions with $\partial {\mathcal {Q}}$. The associated billiard map, which we denote by T, is the Poincaré map under the flow with respect to $\partial {\mathcal {Q}}$.

Parametrizing $\partial O_i$ for each i by arclength r (oriented clockwise), and letting $\varphi \in [-\pi /2, \pi /2]$ denote the angle made by the post-collision velocity vector and the outward normal to $\partial O_i$ at the point of collision, we represent the phase space of our billiard map as $M = \bigcup _{i=1}^d (\partial O_i \times [-\pi /2, \pi /2])$. We denote by $\tau (x)$ the flight time from $x = (r,\varphi ) \in M$ to T(x). We assume that the table has finite horizon, i.e. $\sup _{x\in M}\tau (x)<\infty$. The finite horizon assumption together with the fact that the scatterers are disjoint implies that, there exist constants $\tau _{\min }$, $\tau _{\max } \in {\mathbb {R}}$, such that $0< \tau _{\min } \le \tau (x) \le \tau _{\max } < \infty$ for all $x \in M$.

It is well-known that T preserves a smooth invariant probability measure $\mu _{ \text{ SRB }}$, such that

$$ d\mu _{ \text{ SRB }}(x) = \frac{1}{2|\partial Q|} \cos \varphi \, dr d\varphi . $$

Yet T has discontinuities created by grazing collisions. Set ${\mathcal {S}}_0 = \left\{ (r,\varphi ) \in M: \varphi = {\pm } \frac{\pi }{2} \right\}$ and denote by ${\mathcal {S}}_n = \bigcup _{i=0}^n T^{-i}{\mathcal {S}}_0$ and ${\mathcal {S}}_{-n} = \bigcup _{i=0}^n T^i {\mathcal {S}}_0$ the singularity sets for $T^n$ and $T^{-n}$, respectively. Then $T^n$ is a $C^2$ diffeomorphism of $M {\setminus } {\mathcal {S}}_n$ onto $M {\setminus } {\mathcal {S}}_{-n}$. Note that in this notation, ${\mathcal {S}}_1$ is precisely the same as the set ${\mathcal {S}}$ discussed in Sect. 1.

2.1 Strategy of proof

The class of billiards tables ${\mathcal {Q}}$ we will consider admits the following type of periodic orbit with a grazing collision. If we choose a direction and follow the orbit around one full cycle, then either all grazing collisions along the orbit occur on our right, or all grazing collisions along the orbit occur on our left. See Fig. 1a for an example of an orbit that satisfies our assumption, and Fig. 1b for an example of an orbit that does not. As we will see in the next section, this assumption allows us to easily understand how a neighborhood in ${\mathcal {S}}_0$ of the periodic point is cut under iterations.

Fig. 1

a A periodic orbit of period 4 with grazing collision at $r_0$. The corresponding periodic point is $x_0=(r_0,\pi /2)$. Since there is only one grazing collision, this configuration satisfies our assumption. b A periodic orbit of period 4 with two grazing collisions. It does not satisfy our assumption. Both figures are local and are not meant to illustrate the finite horizon condition

The proofs of Theorems 1.1 and 1.2 will both use the existence of a Cantor rectangle defined in a one-sided neighborhood of the periodic point $x_0 \in {\mathcal {S}}_0$, and it will be divided into the following steps:

• 1. The point $x_0$ has a local stable manifold $W^s(x_0)$ and a local unstable manifold $W^u(x_0)$ of positive length that terminate at $x_0$. Both manifolds are $C^{1+\textrm{Lip}}$, and the wedge between $W^s(x_0)$ and $W^u(x_0)$ belongs to a single component of $M\setminus ({\mathcal {S}}_p \cup {\mathcal {S}}_{-p})$, where p is the period of the orbit.

• 2. Construct a Cantor rectangle ${\mathcal {R}}$ formed by intersections of local stable and local unstable manifolds, with $W^s(x_0)$ and $W^u(x_0)$ comprising two boundaries of ${\mathcal {R}}$, and such that $\mu _{ \text{ SRB }}({\mathcal {R}}) > 0$.

• 3. Use the product structure of the Cantor set and the mixing property of $\mu _{ \text{ SRB }}$ to define a horseshoe K on ${\mathcal {R}}$.

• 4. Use the coding provided by the horseshoe K to construct periodic orbits with large period and arbitrarily large Lyapunov exponent.

• 5. Use the coding for the induced map on the horseshoe K to define a Bernoulli measure whose projection to M is nonadapted.

One of the difficulties involved in the construction is that $DT(x_0)$ is not defined, and DT(x) converges to infinity as x converges to $x_0$. This means that T has huge expansion near $x_0$, and it prevents the use of classical techniques, such as the graph transform method. Nevertheless, the assumption on the periodic orbit allows us to bypass this issue.

Notation. Throughout the text, we will use the notation $A \approx B$ if there exists a constant $C>0$, depending only on the billiard table and the definition of the homogeneity strips (defined in (3.3)), such that $C^{-1} A \le B \le C A$.

3 Construction of the Cantor Rectangle ${\mathcal {R}}$

For ease of the presentation, we assume that the configuration of $x_0=(r_0,\pi /2)$ is the one depicted in Fig. 1a, i.e. $T^4(x_0)=x_0$ with a single grazing collision in one full cycle.³ Let ${\mathcal {K}}$ denote the curvature function of the obstacles. We recall from [CM, Section 4.5] that T has invariant cones:

$$\begin{aligned} \begin{array}{l} {\mathcal {C}}^s_x=\{(dr,d\varphi )\in T_xM: -{\mathcal {K}}-\cos \varphi /\tau \le d\varphi /dr\le -{\mathcal {K}}\}\\ \\ {\mathcal {C}}^u_x=\{(dr,d\varphi )\in T_xM: {\mathcal {K}}\le d\varphi /dr\le {\mathcal {K}}+\cos \varphi /\tau \}. \end{array} \end{aligned}$$

(3.1)

Since the table has finite horizon and the obstacles are disjoint and $C^3$ with strictly positive curvature, these cones are uniformly transverse and bounded away from the lines $r={\textrm{const}}$ and $\varphi ={\textrm{const}}$. According to [CM, Eq. (4.19)], vectors in these cones contract and expand uniformly: there exists $C_e>0$ such that

$$\begin{aligned} \Vert DT^n(x) \vec {v} \Vert \ge C_e \Lambda ^n \Vert \vec {v}\Vert \, \text { for all } \vec {v} \in {\mathcal {C}}^u_x \text { and all }n\ge 1 , \end{aligned}$$

(3.2)

where $\Lambda = 1 + 2 {\mathcal {K}}_{\min } \tau _{\min }$. Similar bounds hold for $\vec {v} \in {\mathcal {C}}^s_x$.

We call a $C^1$ curve W a stable curve if the tangent vector at each $x \in W$ lies in ${\mathcal {C}}^s_x$. Unstable curves are defined similarly.

3.1 Existence of stable and unstable manifolds for $x_0$

In this section, we will construct one-sided stable and unstable manifolds for $x_0$. The main result of this section is the following proposition.

Proposition 3.1

Let $x_0 \in {\mathcal {S}}_0$ be a periodic point with grazing collision as described above. Then $x_0$ has one-sided local stable and unstable manifolds, which are $C^{1+\textrm{Lip}}$ curves in M.

Proposition 3.1 follows from Lemma 3.3.

Let $O_1$ be the obstacle of the grazing collision and let $M_1$ denote the corresponding component of the phase space. We assume that the orientation of the trajectory is counter-clockwise, and that the parametrization of the obstacles is clockwise. Consider a neighborhood $I=[r_1,r_2]\times \{\pi /2\}\subset {\mathcal {S}}_0$ of $x_0$ and let $I_-=[r_1,r_0]\times \{\pi /2\}$ and $I_+=(r_0,r_2]\times \{\pi /2\}$. Since ${\mathcal {S}}_0$ is a dispersing wave front, we may choose $r_1$ and $r_2$ sufficiently close to $r_0$ such that the following behavior holds for the first four iterates of I:

• If $x\in I_- \setminus \{ x_0 \}$ then $r(T(x))>r(T(x_0))$, $r(T^2(x))<r(T^2(x_0))$, $r(T^3(x))>r(T^3(x_0))$, so that $T^4(x)$ hits $O_1$ with $r(T^4(x))<r(T^4(x_0))=r_0$.

• If $x\in I_+$ then $r(T(x))<r(T(x_0))$, $r(T^2(x))>r(T^2(x_0))$, $r(T^3(x))<r(T^3(x_0))$, so that $T^4(x)$ does not hit $O_1$ in a neighborhood $N_\varepsilon (x_0)$ of $x_0$.

This means that I gets cut after four iterations: $T^4(I_-)$ is a curve in $M_1$ terminating at $x_0$, and $T^4(I_+)\cap N_\varepsilon (x_0) = \emptyset$ for some $\varepsilon >0$. Analogous facts hold for $I_+$ under $T^{-1}$. Thus we may choose $r_1$ and $r_2$ sufficiently close to $r_0$ so that $T^4$ is continuous on $I_-$ and $T^{-4}$ is continuous on $I_+$. See Fig. 2.

Fig. 2

A $\delta _0$-neighborhood of $x_0$

Consider the sector ${\mathcal {A}}$ defined by rotating the horizontal segment $I_- = \{ (r, \varphi ): \varphi = \pi /2, r \in [r_1, r_0] \}$ around $x_0$ by an angle of $\pi /2$ radians, until $I_-$ reaches the vertical segment $V = \{ (r, \varphi ): r = r_0, \varphi \in [\pi /2 - (r_0-r_1), \pi /2] \}$. Both segments $I_-$ and V terminate on $x_0$, a fixed point for $T^4$. Moreover, both $T^4(I_-)$ and $T^4(V)$ are unstable curves, tangent at $x_0$ (the cone at $x_0$ consists of a single line with slope ${\mathcal {K}}(r_0)$, see (3.1)), and contained strictly between $I_-$ and V. Since T is orientation preserving, $T^4(I_-)$ comprises the upper boundary of $T^4({\mathcal {A}})$ and $T^4(V)$ comprises its lower boundary. Repeating this argument, since the sector ${\mathcal {A}}$ is strictly contracted by $T^4$, the intersection $T^8({\mathcal {A}}) \cap \mathcal A$, which has upper boundary $T^8(I_-) \cap {\mathcal {A}}$ and lower boundary $T^8(V) \cap {\mathcal {A}}$, lies strictly inside the region with upper boundary $T^4(I_-)$ and lower boundary $T^4(V)$.

Inductively, we see that for each $n \ge 1$ the curve $T^{4n}(I_-)$ contains an unstable curve $\gamma _n(x_0)$, defined by an equation $\{(r,\varphi _n(r)):r\in [r_1,r_0]\}$ where $\varphi _n:[r_1,r_0]\rightarrow {\mathbb {R}}$ is an increasing $C^2$ function with $\varphi _n(r_0)=\pi /2$ and $\varphi _n'(r_0)={\mathcal {K}}(r_0)$. Moreover, the sequence $(\varphi _n)$ is decreasing, and $(\varphi _n')$ is bounded since it must belong to the unstable cone for each n. Thus, $(\varphi _n)$ converges to a function $\varphi ^u:[r_1,r_0]\rightarrow {\mathbb {R}}$. This function defines an unstable curve, which indeed is the local unstable manifold at $x_0$, as we will prove below. We need the following auxiliary lemma.

Lemma 3.2

Suppose $n \ge 1$ and W is an unstable curve passing through $x_0$ such that $T^{-4k}W$ is an unstable curve for all $1 \le k \le n$. The following holds for all $z \in W$:

• (a) $d(T^{-4}z,x_0)\approx d(z,x_0)^2$;

• (b) $d(T^{-4n} z, x_0) \le C_0 d(z, x_0)^{(3/2)^n}$, for some $C_0>0$ independent of W and n.

The analogous statement holds for stable curves W passing through $x_0$ such that $T^{4k}W$ is a stable curve for all $1 \le k \le n$.

This means that near $x_0$, the map $T^4$ exhibits a huge expansion along unstable curves.

Proof

We consider the homogeneity strips

$$\begin{aligned} H_k=\left\{ x=(r,\varphi )\in M: \varphi \in \left[ \tfrac{\pi }{2}-k^{-2},\tfrac{\pi }{2}-(k+1)^{-2}\right) \right\} ,\ k\ge 1. \end{aligned}$$

(3.3)

Notice that $\cos \varphi (x)\approx k^{-2}$ for $x\in H_k$. Let z and W be as in the statement of the lemma and suppose $z \in H_j$. Let V denote the subcurve of W with endpoints z and $x_0$, and write $V=\bigcup _{k\ge j}V_k$, where $V_k=V \cap H_k$. Since $V_k$ is uniformly transversal to the boundaries of $H_k$, we have $|V_k|\approx k^{-3}$, hence

$$ |V|=\sum _{k\ge j} |V_k|\approx \sum _{k\ge j}k^{-3}\approx j^{-2}. $$

We will show that $|T^{-4}V|\approx j^{-4}$, which will complete the proof of part (a).

By [CM, Equation (4.20)], for all $x \in H_k$ we have

$$ J^uT^4(T^{-4}x) \approx \frac{1}{\cos \varphi (x)} \approx k^{2}. $$

Hence $|T^{-4}V_k|\approx |V_k|\cdot k^{-2}\approx k^{-5}$ and so

$$ |T^{-4}V|=\sum _{k\ge j}|T^{-4}V_k|\approx \sum _{k\ge j}k^{-5}\approx j^{-4}, $$

which concludes the proof of part (a).

To prove (b), note that (a) yields a constant $C > 0$ such that $d(T^{-4}z, x_0) \le C d(z, x_0)^2$. Without loss of generality, we may assume that $d(z, x_0) \le C^{-2}$, as this must occur in a fixed number of iterates due to the uniform expansion given by (3.2). Statement (b) then follows directly by iterating (a) since for such z we have $C d( T^{-4k}z, x_0)^2 \le d(T^{-4k}z, x_0)^{3/2}$. $\square$

Lemma 3.3

The curve

$$ W^u(x_0):=\{(r,\varphi ^u(r)):r\in [r_1,r_0]\} $$

is the local unstable manifold at $x_0$ for $T^4$, i.e. $d(T^{-4n}x,x_0)\xrightarrow []{n\rightarrow \infty } 0$ for every $x\in W^u(x_0)$. In particular, $W^u(x_0)$ is a $C^{1+\textrm{Lip}}$ curve.

By time reversibility of billiard systems, $x_0$ also has a $C^{1+\textrm{Lip}}$ local stable manifold $W^s(x_0)$, see Fig. 2.

Proof

It is clear that the function $\varphi ^u$ is Lipschitz and that $W^u(x_0)$ is an unstable curve. Fix $x\in W^u(x_0)$, and consider a stable curve W passing through x. Fix $n\ge 1$. For $1 \le k \le n$, recall that $\gamma _k \subset T^{4k}I_-$ denotes the graph of $\varphi _k$ over the interval $[r_1, r_0]$. Let y be the intersection point of W and $\gamma _n$. Consider the pre-images $T^{-4n}x$ and $T^{-4n}y$, and consider the triangle $\alpha _1\alpha _2\alpha _3$ formed by the three curves:

$$\begin{aligned} \alpha _1&= \text {segment of }{\mathcal {S}}_0\text { joining }x_0\text { and }T^{-4n}y\\ \alpha _2&= \text {curve of }T^{-4n}W\text { joining }T^{-4n}y\text { and }T^{-4n}x\\ \alpha _3&= \text {curve of }W^u(x_0)\text { joining }T^{-4n}x\text { and }x_0. \end{aligned}$$

Since $\alpha _2$ is a stable curve and $\alpha _3$ is an unstable curve, and cones make angles bounded away from zero with respect to horizontal and vertical curves of M, it follows that $d(T^{-4n}x,x_0)\le {\textrm{const}}\times d(T^{-4n}y,x_0)$. Since $\gamma _k$ is an unstable curve for each $k \ge 1$ and $T^{4(k-n)}y \in \gamma _k$, by Lemma 3.2(b) we have

$$ d(T^{-4n}y,x_0) \le C d(T^{-4(n-1)}y,x_0) \le CC_0 d(y,x_0)^{(3/2)^{n-1}}, $$

and so $d(T^{-4n}x,x_0)\le {\textrm{const}}\times d(y,x_0)^{(3/2)^{n-1}}$, which converges to 0 as $n\rightarrow \infty$.

It remains to prove that $\varphi ^u$ is $C^{1+\textrm{Lip}}$. According to the first part of the proof, for each $x\in W^u(x_0)\backslash \{x_0\}$, the curve $W^u(x_0)$ is the unstable manifold of x. By [CM, Corollary 4.61], $W^u(x_0)$ is $C^1$ and its derivative is Lipschitz. $\square$

3.2 The Cantor rectangle ${\mathcal {R}}$

In the last section, we showed the existence of $C^{1+\textrm{Lip}}$ local stable/unstable manifolds $W^{s/u}(x_0)$ at $x_0$ for the map $T^4$. Note that, since these are a stable/unstable curve respectively, they make an angle at $x_0$ that is in⁴$(0,\pi )$. We denote by $\Delta$ the region between these two curves, see Fig. 2. In this section, we will construct a Cantor rectangle of stable and unstable manifolds with $x_0$ as one corner point of the rectangle.

First we recall some terminology. We call D a solid rectangle in M if $\partial D$ comprises four smooth curves, two local stable and two local unstable manifolds. Given a solid rectangle D, let ${\mathfrak {S}}^s(D)$ and ${\mathfrak {S}}^u(D)$ denote the set of local stable and unstable manifolds, respectively, of points in D that do not terminate in the interior of D. Define the Cantor rectangle

$$\begin{aligned} R(D) = {\mathfrak {S}}^s(D) \cap {\mathfrak {S}}^u(D) \cap D. \end{aligned}$$

By construction, R(D) has a locally maximal hyperbolic product structure. We will work with such Cantor rectangles R, closed sets formed by locally maximal intersections of stable and unstable manifolds such that Leb$(R)>0$. Conversely, given such a Cantor rectangle R, we denote by D(R) the smallest solid rectangle containing R.

Unlike [CM, Section 7.11], we do not restrict ourselves to H–manifolds, i.e. to local stable (respectively unstable) manifolds whose forward (respectively backward) trajectories lie in a single homogeneity strip at each step. We do this because the rectangle we construct, ${\mathcal {R}}$, will necessarily cross infinitely many homogeneity strips in order to incorporate $x_0$ in its boundary.

Now consider a segment of $W^u(x_0)$ that lies between homogeneity strips of index $k_1$ and $k_1^2$. Call this curve $A_1$. Since $A_1$ is a finite union of homogeneous unstable manifolds, almost every point on $A_1$ has stable manifolds of positive length [CM, Theorem 5.70].

Now consider $T^{-4}A_1 \subset W^u(x_0)$. If W is a local stable manifold of a point in $A_1$, then $T^{-4}W$ is a local stable manifold of a point in $T^{-4}A_1$, and it has one connected component intersecting $\Delta$. This component cannot be cut until it exits a $\delta _0$–neighborhood of $x_0$, and it cannot cross $W^s(x_0)$ (since stable manifolds do not intersect). Therefore,

$$\begin{aligned} |W^s(y)| \ge \min \left\{ \frac{|W^s(T^4y)|}{ J^sT^4(y)}, \delta _0 \right\} \, \text{ for } \text{ all } y \in T^{-4}A_1. \end{aligned}$$

Note that $J^sT^4(y) \approx \cos \varphi (y)$ so that the expansion is large in a $\delta _0$–neighborhood of $x_0$.

Applying this construction inductively, we are guaranteed the existence of increasingly longer stable manifolds for points in $W^u(x_0)$ as we approach $x_0$, and so a positive measure of such manifolds extend a minimum distance of $\delta _1 > 0$ inside $\Delta$.

By time-reversal symmetry, this construction also implies that almost every point on $W^s(x_0)$ has an unstable manifold of positive length, and that unstable manifolds of a minimum length $\delta _1$ comprise a positive measure set of points inside $W^s(x_0)$.

Fixing once and for all $\delta _0,\delta _1 >0$ small enough such that the above constructions hold in the $\delta _0$–neighborhood of $x_0$, we obtain a (locally maximal) closed, Cantor rectangle ${\mathcal {R}}$ comprised of the intersection of a positive measure set of stable and unstable manifolds, having a segment of $W^s(x_0)$ along one boundary, a segment of $W^u(x_0)$ along another, and containing the point $x_0$ as one of its corner points.

3.3 Definition of the horseshoe

We let $G=T^4$, and construct a horseshoe for a power of G. The idea is to have two branches for the horseshoe, which are two stable rectangles in ${\mathcal {R}}$, one of them containing $x_0$. We first show that an iterate of $W^u(x_0)$ crosses $W^s(x_0)$, i.e. there is an homoclinic intersection. This is a consequence of standard results in the theory of dispersing billiards.

We need a little more terminology. Given a Cantor rectangle R, we call $S \subset R$ an s–subrectangle of R if for each $x \in S$, $W^s(x) \cap S = W^s(x) \cap R$. Similarly, $U \subset R$ is a u–subrectangle if $W^u(x) \cap U = W^u(x) \cap R$ for all $x \in U$. We say a local unstable manifold $W^u$ fully crosses R if $W^u \cap \mathring{D}(R) \ne \emptyset$ and $W^u$ does not terminate in the interior of D(R). A Cantor rectangle $R'$ u–crosses R if every unstable manifold $W^u \in {\mathfrak {S}}^u(R')$ fully crosses R. Analogous definitions hold for stable manifolds and s–crossings.

Given a Cantor rectangle R, $T^n(R)$ is a finite union of (maximal) Cantor rectangles (recall that stable manifolds cannot be cut by singularities of T). We label them $R_{n,i}$. Then each $T^{-n}(R_{n,i})$ is an s–subrectangle of R.

With these preliminaries, we are ready to proceed with the construction of our horseshoe.

• By [CM, Lemma 7.87], there is a finite collection of rectangles $R_1,\ldots ,R_N$ of positive measure each such that any stable and unstable curve of length at least $\delta _1/2$ properly crosses at least one of the rectangles. Without loss of generality, we can assume that $W^u(x_0)$ properly crosses $R_1$ and $W^s(x_0)$ properly crosses $R_2$.

• By [CM, Lemma 7.90], there is a ‘magnet’ rectangle $R^*$ of positive measure, and a ‘high density’ subset ${\mathfrak {P}}^* \subset R^*$, satisfying the following property: if $R_{k,n,i}$ is a maximal rectangle in $T^n(R_k)$ and $R_{k,n,i} \cap {\mathfrak {P}}^* \ne \emptyset$ where n is large enough, then $R_{k,n,i}$ u–crosses $R^*$.⁵ The analogous properties hold for maximal rectangles $R_{k,-n,i}$ of $T^{-n}(R_k)$ and s–crossings of $R^*$.

• Since $\mu _{ \text{ SRB }}$ is mixing, there are $m,n>0$ such that $G^n(R_1)\cap {\mathfrak {P}}^*\ne \emptyset$ and $G^{-m}(R_2)\cap {\mathfrak {P}}^*\ne \emptyset$. Therefore there are an s-subrectangle $R'\subset R_1$ and a u-subrectangle $R''\subset R_2$ such that $G^n(R')$ u-crosses $R^*$ and $G^{-m}(R'')$ s-crosses $R^*$. This implies that $G^n(W^u(x_0))\cap G^{-m}(W^s(x_0))\ne \emptyset$, proving the desired transverse intersection.

Let $x_1\in W^u(x_0)\cap G^{-(m+n)}W^s(x_0)$. Take points $z\in W^u(x_0)\cap {\mathcal {R}}$ and $w\in W^s(x_0)\cap {\mathcal {R}}$ such that $x_1$ belongs to the segment of $W^u(x_0)$ joining $x_0$ and z. The points $x_0,z,[z,w],w$ define a solid rectangle D. Let $U_0,U_1$ be disjoint s-subrectangles of D such that $U_0$ contains $x_0$ as a vertex and the solid rectangle of $U_1$ contains $x_1$ in the interior of its u–side. If $\ell >0$ is large, then $G^\ell (U_0)$ u-crosses D. Also, by construction, $G^{m+n}(U_1)$ intersects $W^s(x_0)$ transversally, and so if $\ell _0>0$ is large enough then $G^{\ell _0+m+n}(U_1)$ also u-crosses D. Hence, we can fix $\ell _0>0$ such that for $\ell =\ell _0+m+n$ the sets $G^\ell (U_0),G^\ell (U_1)$ both u-cross D. Letting $f=G^{\ell }$, we obtain a horseshoe for f as the intersection

$$\begin{aligned} K:= \bigcap _{n=-\infty }^\infty f^n(U_0 \cup U_1) \,. \end{aligned}$$

By construction, K is conjugate to a full shift in two symbols, i.e. if $(\Sigma ,\sigma )$ is the topological Markov shift with $\Sigma =\{0,1\}^{{\mathbb {Z}}}$, then there is a measurable bijection $\pi :\Sigma \rightarrow K$ such that $\pi \circ \sigma =f\circ \pi$.

4 Proof of Theorem 1.1

The idea to prove Theorem 1.1 is the following: points that spend many iterates near $x_0$ approach $x_0$ super-exponentially fast. This is expressed for unstable curves (and analogously for stable curves) by Lemma 3.2. Given $x\in \Delta$, let $W^s,W^u$ be a stable, unstable curve passing through x such that $\{x_s\}=W^u\cap W^s(x_0)$ and $\{x_u\}=W^s\cap W^u(x_0)$ are defined. Hence

$$ d(x,x_0)\approx \max \{ d(x_s,x_0),d(x_u,x_0)\}. $$

Denote an element of $\Sigma$ by ${\underline{v}}=(v_n)_{n\in {\mathbb {Z}}}$. Let $C=\{{\underline{v}}\in \Sigma :v_0=1\}$, $C_n=\{{\underline{v}}\in C:v_1=\cdots =v_{n-1}=0\text { and }v_n=1\}$ for $n\ge 1$, and $C_\infty =\{{\underline{v}}\in C:v_n=0\text { for all }n\ge 1\}$. Clearly $C {\setminus } C_\infty =\bigcup _{n\ge 1}C_n$, and the first return map ${{\widetilde{\sigma }}}:C\rightarrow C$ of $\sigma$ to C is a full topological Markov shift in an infinite countable alphabet with $\widetilde{\sigma }=\sigma ^n$ on $C_n$. For $n \ge 1$, let $x=\pi ({\underline{v}})$ with ${\underline{v}}\in C_n$.

As in the previous paragraph, let $W^u$ be an unstable curve connecting f(x) and $W^s(x_0)$ and define $\{ x^1_s \} = W^u \cap W^s(x_0)$. Let $W^s$ be a stable curve connecting $f^{n-1}(x)$ and $W^u(x_0)$ and define $\{ x^{n-1}_u \} = W^s \cap W^u(x_0)$. Since $f^k(x^1_s) \in W^s(x_0)$ and $f^{-k}(x^{n-1}_u) \in W^u(x_0)$ for $k \ge 0$, we may apply Lemma 3.2(b) and obtain that:

• $d(f^{\lfloor \frac{n}{2}\rfloor }x^1_s,x_0) \le C_0 d(x^1_s,x_0)^{(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}$.

• $d(f^{-\lfloor \frac{n}{2}\rfloor }x^{n-1}_u,x_0) \le C_0 d(x^{n-1}_u,x_0)^{(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}$.

Since $W^s$ is a stable curve, $f^{-\lfloor \frac{n}{2} \rfloor }(W^s)$ is again a stable curve connecting $f^{\lfloor \frac{n}{2} \rfloor }(x)$ with $W^u(x_0)$. Similarly, since $W^u$ is an unstable curve, $f^{\lfloor \frac{n}{2} \rfloor }(W^u)$ is an unstable curve connecting $f^{\lfloor \frac{n}{2} \rfloor }(x)$ with $W^s(x_0)$. In particular, letting $\delta =\textrm{diam}(D)<1$, there is a constant $C \ge 1$ such that

$$\begin{aligned} d(f^{\lfloor \frac{n}{2}\rfloor }x, {\mathcal {S}}_0)\le C\delta ^{(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }},\ \forall x\in \pi (C_n). \end{aligned}$$

(4.1)

Let $b=\sum _{n\ge 1} \left( \frac{3}{2}\right) ^{-{\lfloor \frac{n}{2}\rfloor }}<\infty$. Take $p_n=b^{-1} \left( \frac{3}{2}\right) ^{-{\lfloor \frac{n}{2}\rfloor }}$ for $n\ge 1$, and consider a Bernoulli probability measure ${{\widetilde{\nu }}}$ on $C\cong {\mathbb {N}}^{{\mathbb {Z}}}$ such that ${{\widetilde{\nu }}}(C_n)=p_n$. This measure induces a $\sigma$–invariant probability measure $\nu$ on $\Sigma$, which descends by $\pi$ to a f–invariant probability measure $\eta$ on K. Note that $\nu$ is supported on the disjoint union $\bigcup _{n\ge 1}\bigcup _{0\le k<n}\sigma ^{k}C_n$, and $\eta$ is supported on the disjoint union $\bigcup _{n\ge 1}\bigcup _{0\le k<n}f^{k}\pi (C_n)$. Applying (4.1),

$$\begin{aligned} \int |\log d(x, {\mathcal {S}}_0)| \, d\eta&\ge \int _{\bigcup _{n\ge 1}f^{\lfloor \frac{n}{2}\rfloor }\pi (C_n)} | \log d(x, {\mathcal {S}}_0) | \, d\eta \\&= \sum \limits _{n \ge 1} \int _{\pi (C_n)} | \log d(f^{\lfloor \frac{n}{2}\rfloor }x, {\mathcal {S}}_0) | \, d\eta \\&\ge \sum _{n\ge 1} \left| \log \left( C\delta ^{(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}\right) \right| \eta [\pi (C_n)] \\&\ge - \log C+ |\log \delta | \sum _{n\ge 1}p_n \left( \tfrac{3}{2}\right) ^{\lfloor \frac{n}{2}\rfloor }=\infty . \end{aligned}$$

Now we estimate $h_\eta (f)$. Let n be the return time function defining ${{\widetilde{\sigma }}}$. By the Abramov formula, and using that ${{\widetilde{\nu }}}$ is a Bernoulli measure, we have

$$\begin{aligned} h_\eta (f)= & {} h_\nu (\sigma )=\frac{h_{{{\widetilde{\nu }}}}(\widetilde{\sigma })}{\int _C n({\underline{v}})d{{\widetilde{\nu }}}({\underline{v}})} =\frac{\displaystyle \sum _{n=1}^\infty - p_n \log p_n}{\displaystyle \sum _{n=1}^\infty n p_n } \\\approx & {} \frac{\displaystyle \sum _{n=1}^\infty \left( \tfrac{3}{2}\right) ^{-\lfloor \frac{n}{2}\rfloor }{\left\lfloor \tfrac{n}{2}\right\rfloor } \log (3/2)}{\displaystyle \sum _{n=1}^\infty n \left( \tfrac{3}{2}\right) ^{-\lfloor \frac{n}{2}\rfloor } } \approx \frac{\log (3/2)}{2}>0. \end{aligned}$$

Finally, let $\mu =\frac{1}{4\ell }\sum \limits _{k=0}^{4\ell -1}\eta \circ T^{-k}$, which is a T–invariant probability measure. We claim that $\mu$ is a measure satisfying Theorem 1.1. Firstly, we have

$$\begin{aligned} \int | \log d(x, {\mathcal {S}}_0) | \, d\mu= & {} \frac{1}{4\ell }\sum _{k=0}^{4\ell -1}\int | \log d(x, {\mathcal {S}}_0) | \, d(\eta \circ T^{-k})\nonumber \\\ge & {} \frac{1}{4\ell } \int | \log d(x, {\mathcal {S}}_0) | \, d\eta =\infty . \end{aligned}$$

(4.2)

Since ${\mathcal {S}}_0 \subset {\mathcal {S}}_1$, this implies that the measure $\mu$ is not adapted.

Now, using that each $\eta \circ T^{-k}$ is f–invariant, it follows again by the Abramov formula and the linearity of the metric entropy that

$$ h_\mu (T)=\frac{1}{4\ell }h_\mu (f)=\frac{1}{16\ell ^2}\sum _{k=0}^{4\ell -1}h_{\eta \circ T^{-k}}(f)\ge \frac{\log (3/2)}{32\ell ^2}>0. $$

This concludes the proof of Theorem 1.1.

5 Proof of Theorem 1.2

The subset $\pi (C_n)$ contains a periodic orbit of period n for f, equal to $y_n=\pi ({\underline{v}})$ where $v_0=1$ and $v_1=\cdots =v_{n-1}=0$. Due to (4.1), we have $d(f^{\lfloor \frac{n}{2}\rfloor }y_n, {\mathcal {S}}_0)\le C\delta ^{(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}$ and so, using again [CM, eq. 4.20],

$$\begin{aligned} J^uf(f^{\lfloor \frac{n}{2}\rfloor -1}y_n) \ge (\cos \varphi (f^{\lfloor \frac{n}{2} \rfloor } y_n))^{-1} \approx d(f^{\lfloor \frac{n}{2}\rfloor }y_n, {\mathcal {S}}_0)^{-1} \ge C^{-1} \delta ^{-(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}. \end{aligned}$$

The Lyapunov exponent for the point $y_n$ with respect to f is at least

$$\begin{aligned} \begin{aligned} \lim _{j \rightarrow \infty } \frac{1}{jn} \log J^uf^{jn}(y_n)&\ge \lim _{j \rightarrow \infty } \frac{1}{jn} j \log J^uf(f^{\lfloor \frac{n}{2}\rfloor -1 }y_n) \\&\ge \frac{1}{n} \log \left( C^{-1} \delta ^{-(3/2)^{\left\lfloor \frac{n}{2}\right\rfloor }}\right) \approx \frac{\left( \tfrac{3}{2}\right) ^{\lfloor \frac{n}{2}\rfloor }|\log \delta |}{n}. \end{aligned} \end{aligned}$$

Therefore, the Lyapunov exponent for the point $y_n$ with respect to T is at least of the order of $\left( \frac{3}{2}\right) ^{\lfloor \frac{n}{2}\rfloor }/ 4\ell n$. We have thus constructed a sequence of periodic orbits with finite, but arbitrarily large Lyapunov exponents. This proves the first statement of Theorem 1.2.

Equation (1.3) follows immediately since each of the above periodic orbits contains no grazing collisions and therefore each corresponding invariant measure is supported outside of an $\varepsilon$–neighborhood of ${\mathcal {S}}_0$ for some $\varepsilon >0$.

To prove the final statement of the theorem, equation (1.4), let $\mu$ be the T–invariant measure constructed in the proof of Theorem 1.1. Using again [CM, (4.20)], or [CM, eq. (5.36)], we have $J^uT(x) \approx (\cos \varphi (Tx))^{-1}$. By the invariance of $\mu$ and estimate (4.2), we get that

$$\begin{aligned} \int \log J^uT \, d\mu\approx & {} \int - \log \cos (\varphi \circ T) \, d\mu \\= & {} \int - \log \cos \varphi \, d\mu \ge \int |\log d(x, {\mathcal {S}}_0)| \, d\mu (x) = \infty \,. \end{aligned}$$

Since $h_\mu (T) >0$, this proves (1.4).