Abstract
For any polynomial diffeomorphism f of \({\mathbb C}^2\) with positive entropy, the Julia set of f is never \(C^1\) smooth as a manifold-with-boundary.
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1 Introduction
There are several reasons why the polynomial diffeomorphisms of \({\mathbb C}^2\) form an interesting family of dynamical systems. One of these is the fact that there are connections with two other areas of dynamics: polynomial maps of \({\mathbb C}\) and diffeomorphisms of \({\mathbb R}^2\), which have each received a great deal of attention. Among the endomorphisms of \({\mathbb P}^k\), certain ones have more special, and regular, geometric structure.
The question arises whether, among the polynomial diffeomorphisms of \({\mathbb C}^2\), are there analogous special maps with special geometry? The Julia set of such a special map would be expected to have some smoothness. Here we show that this does not happen.
More generally, we consider a holomorphic mapping \(f:X\rightarrow X\) of a complex manifold X. The Fatou set of f is defined as the set of points \(x\in X\) where the iterates \(f^n:=f\circ \cdots \circ f\) are locally equicontinuous. If X is not compact, then in the definition of equicontinuity, we consider the one point compactification of X; in this case, a sequence which diverges uniformly to infinity is equicontinuous. By the nature of equicontinuity, the dynamics of f is regular on the Fatou set. The Julia set is defined as the complement of the Fatou set, and this is where any chaotic dynamics of f will take place. The first nontrivial case is where \(X={\mathbb P}^1\) is the Riemann sphere, and in this case Fatou (see [17]) showed that if the Julia set J is a smooth curve, then either J is the unit circle or J is a real interval. If J is the circle, then f is equivalent to \(z\mapsto z^d\), where d is an integer with \(|d|\ge 2\); if J is the interval, then f is equivalent to a Chebyshev polynomial. These maps with smooth J play special roles, and this sparked our interest to look for smooth Julia sets in other cases. (The higher dimensional case is discussed, for instance, in Nakane [19] and Uchimura [23, 24].)
Here we address the case where \(X={\mathbb C}^2\), and f is a polynomial automorphism, which means that f is biholomorphic, and the coordinates are polynomials. Since f is invertible, there are two Julia sets: \(J^+\) for iterates in forward time, and \(J^-\) for iterates in backward time. Polynomial automorphisms have been classified by Friedland and Milnor [12]; every such automorphism is conjugate to a map which is either affine or elementary, or it belongs to the family \(\mathcal{H}\). The affine and elementary maps have simple dynamics, and \(J^\pm \) are (possibly empty) algebraic sets (see [12]).
Thus we will restrict our attention to the maps in \(\mathcal{H}\), which are finite compositions \(f:=f_k\circ \cdots \circ f_1\), where each \(f_j\) is a generalized Hénon map, which by definition has the form \(f_j(x,y)=(y, p_j(y)-\delta _j x)\), where \(\delta _j\in {\mathbb C}\) is nonzero, and \(p_j(y)\) is a monic polynomial of degree \(d_j\ge 2\). The degree of f is \(d:=d_1\cdots d_k\), and the complex Jacobian of f is \(\delta :=\delta _1\cdots \delta _k\). In [12] and [22], it is shown that the topological entropy of f is \(\log d>0\). The dynamics of such maps is complicated and has received much study, starting with the papers [3, 11, 14, 15].
For maps in \(\mathcal{H}\), we can ask whether \(J^+\) can be a manifold. For any saddle point q, the stable manifold \(W^s(q)\) is a Riemann surface contained in \(J^+\). Thus \(J^+\) would have to have real dimension at least two. However, \(J^+\) is also the support of a positive, closed current \(\mu ^+\) with continuous potential, and such potentials cannot be supported on a Riemann surface (see [3, 11]). On the other hand, since \(J^+=\partial K^+\) is a boundary, it cannot have interior. Thus dimension 3 is the only possibility for \(J^+\) to be a manifold. In fact, there are examples of f for which \(J^+\) has been shown to be a topological 3-manifold (see [8, 11, 16, 20]). Fornæss and Sibony [11] have shown that \(J^+\) cannot be smooth for a generic element of \(\mathcal{H}\).
The purpose of this paper is to prove the following:
Theorem
For any polynomial automorphism of \({\mathbb C}^2\) of positive entropy, neither \(J^+\) nor \(J^-\) is smooth of class \(C^1\), in the sense of manifold-with-boundary.
We may interchange the roles of \(J^+\) and \(J^-\) by replacing f by \(f^{-1}\), so there is no loss of generality if we consider only \(J^+\).
In an Appendix, we discuss the nonsmoothness of the related sets J, \(J^*\), and K.
2 No Boundary
Let us start by showing that if \(J^+\) is a \(C^1\) manifold-with-boundary, then the boundary is empty. Recall that if \(J^+\) is \(C^1\), then for each \(q_0\in J^+\) there is a neighborhood \(U\ni q_0\) and \(r,\rho \in C^1(U)\) with \(dr\wedge d\rho \ne 0\) on U, such that \(U\cap J^+=\{r=0, \rho \le 0\}\). If \(J^+\) has boundary, it is given locally by \(\{r=\rho =0\}\). For \(q\in J^+\), the tangent space \(T_qJ^+\) consists of the vectors that annihilate dr. This contains the subspace \(H_q\subset T_qJ^+\) consisting of the vectors that annihilate \(\partial r\). \(H_q\) is the unique complex subspace inside \(T_qJ^+\), so if \(M\subset J^+\) is a complex submanifold, then \(T_qM=H_q\).
We start by showing that if \(J^+\) is \(C^1\), then it carries a Riemann surface lamination.
Lemma 2.1
If \(J^+\) is \(C^1\) smooth, then \(J^+\) carries a Riemann surface foliation \(\mathcal{R}\) with the property that if \(W^s(q)\) is the stable manifold of a saddle point q, then \(W^s(q)\) is a leaf of \(\mathcal{R}\). If \(J^+\) is a \(C^1\) smooth manifold-with-boundary, then \(\mathcal{R}\) extends to a Riemann surface lamination of \(J^+\). In particular, any boundary component is a leaf of \(\mathcal{R}\).
Proof
Given \(q_0\in J^+\), let us choose holomorphic coordinates (z, w) such that \(dr(q_0) = dw\). We work in a small neighborhood which is a bidisk \(\Delta _\eta \times \Delta _\eta \). We may choose \(\eta \) small enough that \(|r_z/r_w|<1\). In the (z, w)-coordinates, the tangent space \(H_q\) has slope less than 1 at every point \(\{|z|,|w|<\eta \}\). Now let \(\hat{q}\) be a saddle point, and let \(W^s(\hat{q})\) be the stable manifold, which is a complex submanifold of \({\mathbb C}^2\), contained in \(J^+\). Let M denote a connected component of \(W^s(\hat{q})\cap (\Delta _\eta \times \Delta _{\eta /2})\). Since the slope is <1, it follows that there is an analytic function \(\varphi :\Delta _\eta \rightarrow \Delta _\eta \) such that \(M\subset \Gamma _\varphi :=\{(z,\varphi (z)):z\in \Delta _\eta \}\). Let \(\Phi \) denote the set of all such functions \(\varphi \). Since a stable manifold can have no self-intersections, it follows that if \(\varphi _1,\varphi _2\in \Phi \), then either \(\Gamma _{\varphi _1}=\Gamma _{\varphi _2}\) or \(\Gamma _{\varphi _1}\cap \Gamma _{\varphi _2}=\emptyset \). Now let \(\hat{\Phi }\) denote the set of all normal limits (uniform on compact subsets of \(\Delta _\eta \)) of elements of \(\Phi \). We note that by Hurwitz’s Theorem, the graphs \(\Gamma _\varphi \), \(\varphi \in \hat{\Phi }\) have the same pairwise disjointness property. Finally, by [4], \(W^s(q_0)\) is dense in \(J^+\), so the graphs \(\Gamma _\varphi \), \(\varphi \in \hat{\Phi }\) give the local Riemann surface lamination.
If \(q_1\) is another saddle point, we may follow the same procedure and obtain a Riemann surface lamination whose graphs are given locally by \(\varphi \in \hat{\Phi }_1\). However, we have seen that the tangent space to the foliation at a point q is given by \(H_q\). Since these two foliations have the same tangent spaces everywhere, they must coincide.
We have seen that all the graphs are contained in \(J^+\), so if \(J^+\) has boundary, then the boundary must coincide locally with one of the graphs. \(\square \)
We will use the observation that \(K^+\subset \{(x,y)\in {\mathbb C}^2:|y|>\max (|x|,R)\}\). Further, we will use the Green function \(G^-\) which has many properties, including
- (i):
-
\(G^-\) is pluriharmonic on \(\{G^->0\}\),
- (ii):
-
\(\{G^-=0\}=K^-\), and
- (iii):
-
\(G^-\circ f = d^{-1}G^-\).
Further, the restriction of \(G^-\) to \(\{|y|\le \max (|x|,R)\}\) is a proper exhaustion.
Lemma 2.2
Suppose that \(J^+\) is a \(C^1\) smooth manifold-with-boundary, and M is a component of the boundary of \(J^+\). Then M is a closed Riemann surface, and \(M\cap K\ne \emptyset \).
Proof
We consider the restriction \(g:=G^-|_{M}\). If \(M\cap K=\emptyset \), then g is harmonic on M. On the other hand, g is a proper exhaustion of M, which means that \(g(z)\rightarrow \infty \) as \(z\in M\) leaves every compact subset of M. This means that g must assume a minimum value at some point of M, which would violate the minimum principle for harmonic functions. \(\square \)
Lemma 2.3
Suppose that \(J^+\) is a \(C^1\) smooth manifold-with-boundary. Then the boundary is empty.
Proof
Let M be a component of the boundary of \(J^+\). By Lemma 2.2, M must intersect \(\Delta _R^2\). Since \(J^+\) is \(C^1\), there can be only finitely many boundary components of \(J^+\cap \Delta _R^2\). Thus there can be only finitely many components M, which must be permuted by f. If we take a sufficiently high iterate \(f^N\), we may assume that M is invariant. Now let \(h:=f^N|_{M}\) denote the restriction to M. We see that h is an automorphism of the Riemann surface M, and the iterates of all points of M approach \(K\cap M\) in forward time. It follows that M must have a fixed point \(q\in M\), and \(|h'(q)|<1\). The other multiplier of Df at q is \(\delta /h'(q)\).
We consider three cases. First, if \(|\delta /h'(q)|>1\), then q is a saddle point, and \(M=W^s(q)\). On the other hand, by [4], the stable manifold of a saddle points is dense in \(J^+\), which makes it impossible for M to be the boundary of \(J^+\). This contradiction means that there can be no boundary component M.
The second case is \(|\delta /h'(q)|<1\). This case cannot occur because the multipliers are less than 1, so q is a sink, which means that q is contained in the interior of \(K^+\) and not in \(J^+\).
The last case is where \(|\delta /h'(q)|=1\). In this case, we know that f preserves \(J^+\), so Df must preserve \(T_q(J^+)\). This means that the outward normal to M inside \(J^+\) is preserved, and thus the second multiplier must be +1. It follows that q is a semi-parabolic/semi-attracting fixed point. It follows that \(J^+\) must have a cusp at q and cannot be \(C^1\) (see Ueda [25] and Hakim [13]). \(\square \)
3 Maps that Do Not Decrease Volume
We note the following topological result (see Samelson [21] for an elegant proof): If M is a smooth 3-manifold (without boundary) of class \(C^1\) in \({\mathbb R}^4\), then it is orientable. This gives:
Proposition 3.1
For any \(q\in M\), there is a neighborhood U about q so that \(U-M\) consists of two components \(\mathcal{O}_1\) and \(\mathcal{O}_2\), which belong to different components of \({\mathbb R}^4-M\).
Proof
Suppose that \(\mathcal{O}_1\) and \(\mathcal{O}_2\) belong to the same component of \({\mathbb R}^4-M\). Then we can construct a simple closed curve \(\gamma \subset {\mathbb R}^4\) which crosses M transversally at q and has no other intersection with M. It follows that the (oriented) intersection is \(\gamma \cdot M=1\) (modulo 2). But the oriented intersection modulo 2 is a homotopy invariant (see [18]), and \(\gamma \) is contractible in \({\mathbb R}^4\), so we must have \(\gamma \cdot M=0\) (modulo 2). \(\square \)
Corollary 3.2
If \(J^+\) is \(C^1\) smooth, then f is an orientation preserving map of \(J^+\).
Proof
\(U^+:={\mathbb C}^2-K^+\) is a connected (see [15]) and thus it is a component of \({\mathbb C}^2-J^+\). Since f preserves \(U^+\), it also preserves the orientation of \(J^+\), which is \(\pm \partial U^+\). \(\square \)
We recall the following result of Friedland and Milnor:
Theorem
([12]) If \(|\delta |>1\), then \(K^+\) has zero Lebesgue volume, and thus \(J^+=K^+\). If \(|\delta |=1\), then \({\mathrm{int}}(K^+)={\mathrm{int}}(K^-) = {\mathrm{int}}(K)\). In particular, there exists R such that \(J^+=K^+\) outside \(\Delta _R^2\).
Proof of Theorem in the case \(|\delta |\ge 1\). Let \(q\in J^+\) be a point outside \(\Delta ^2_R\), as in the Theorem above. Then near q there must be a component \(\mathcal{O}\), which is distinct from \(U^+={\mathbb C}^2-K^+\). Thus \(\mathcal{O}\) must belong to the interior of \(K^+\). But by the Theorem above, the interior of \(K^+\) is not near q. \(\square \)
4 Volume Decreasing Maps
Throughout this section, we continue to suppose that \(J^+\) is \(C^1\) smooth, and in addition we suppose that \(|\delta |<1\). For a point \(q\in J^+\), we let \(T_q:=T_q(J^+)\) denote the real tangent space to \(J^+\). We let \(H_q:=T_q\cap i T_q\) denote the unique (one-dimensional) complex subspace inside \(T_q\). Since \(J^+\) is invariant under f, so is \(H_q\), and we let \(\alpha _q\) denote the multiplier of \(D_qf|_{H_q}\).
Lemma 4.1
Let \(q\in J^+\) be a fixed point. There is a \(D_qf\)-invariant subspace \(E_q\subset T_q({\mathbb C}^2)\) such that \(H_q\) and \(E_q\) generate \(T_q\). We denote the multiplier of \(D_qf|_{E_q}\) by \(\beta _q\). Thus \(D_qf\) is linearly conjugate to the diagonal matrix with diagonal elements \(\alpha _q\) and \(\beta _q\). Further, \(\beta _q\in {\mathbb R}\) and \(\beta _q>0\).
Proof
We have identified an eigenvalue \(\alpha _q\) of \(D_qf\). If \(D_qf\) is not diagonalizable, then it must have a Jordan canonical form \(\left( \begin{array}{cc}\alpha _q &{} 1\\ 0&{} \alpha _q\end{array}\right) \). The determinant is \(\alpha _q^2=\delta \), which has modulus less than 1. Thus \(|\alpha _q|<1\), which means that q is an attracting fixed point and thus in the interior of \(K^+\), not in \(J^+\). Thus \(D_qf\) must be diagonalizable, which means that \(H_q\) has a complementary invariant subspace \(E_q\). Since \(E_q\) and \(T_q\) are invariant under \(D_qf\), the real subspace \(E_q\cap T_q\subset E_q\) is invariant, too. Thus \(\beta _q\in {\mathbb R}\). By Corollary 3.2, \(D_qf\) will preserve the orientation of \(T_q\), and so \(\beta _q>0\). \(\square \)
Let us recall the Riemann surface foliation of \(J^+\) which was obtained in Lemma 2.1. For \(q\in J^+\), we let \(R_q\) denote the leaf of \(\mathcal{R}\) containing q. If q is a fixed point, then f defines an automorphism \(g:=f|_{R_q}\) of the Riemann surface \(R_q\). Since \(R_q\subset K^+\), we know that the iterates of \(g^n\) are bounded in a complex disk \(q\in \Delta _q\subset R_q\). Thus the derivatives \((Dg)^n= D(g^n)\) are bounded at q. We conclude that \(|\alpha _q|=|D_q(g)|\le 1\). If \(|\alpha _q|=1\), then \(\alpha _q\) is not a root of unity. Otherwise g is an automorphism of \(R_q\) fixing q, and \(Dg^n(q) =1\) for some n. It follows that \(g^n\) must be the identity on \(R_q\). This means that Rq would be a curve of fixed points for \(f^n\), but by [FM] all periodic points of f are isolated, so this cannot happen.
Lemma 4.2
If \(q\in J^+\) is a fixed point, then q is a saddle point, and \(\alpha _q=\delta /d\), and \(\beta _q=d\).
Proof
First we claim that \(|\alpha _p|<1\). Otherwise, we have \(|\alpha _q|=1\), and by the discussion above, this means that \(\alpha _q\) is not a root of unity. Thus the restriction \(g = f|_{R_q}\) is an irrational rotation. Let \(\Delta \subset R_q\) denote a g-invariant disk containing q. Since \(|\delta |=|\alpha _q\beta _q|=|\beta _q|\) has modulus less than 1, we conclude that f is normally attracting to \(\Delta \), and thus q must be in the interior of \(K^+\), which contradicts the assumption that \(q\in J^+\).
Now we have \(|\alpha _q|<1\), so if \(|\beta _q|=1\), we have \(\beta _q=1\), since \(\beta _q\) is real and positive. This means that q is a semi-parabolic, semi-attracting fixed point for f. We conclude by Ueda [25] and Hakim [13] that \(J^+\) has a cusp at q and thus is not smooth. Thus we conclude that \(|\beta _q|>1\), which means that q is a saddle point.
Now since \(E_q\) is transverse to \(H_q\), it follows that \(W^u(q)\) intersects \(J^+\) transversally, and thus \(J^+\cap W^u(q)\) is \(C^1\) smooth. Let us consider the uniformization
The pre-image \(\tau :=\phi ^{-1}(W^u(q)\cap J^+)\subset {\mathbb C}\) is a \(C^1\) curve passing through the origin and invariant under \(\zeta \mapsto \lambda ^u\zeta \). It follows that \(\lambda ^u\in {\mathbb R}\), and \(\tau \) is a straight line containing the origin. Further, \(g^+:=G^+\circ \phi \) is harmonic on \({\mathbb C}-\tau \), vanishing on \(\tau \), and satisfying \(g^+(\lambda ^u\zeta )=d\cdot g^+(\zeta )\). Since \(\tau \) is a line, it follows that \(g^+\) is piecewise linear, so we must have \(\lambda ^u=\pm d\). Finally, since f preserves orientation, we have \(\lambda ^u=d\).
Lemma 4.3
There can be at most one fixed point in the interior of \(K^+\). There are at least \(d-1\) fixed points contained in \(J^+\), and at each of these fixed points, the differential Df has multiplier of d.
Proof
Suppose that q is a fixed point in the interior of \(K^+\). Then q is contained in a recurrent Fatou domain \(\Omega \), and by [4], \(\partial \Omega =J^+\). If there is more than one fixed point in the interior of \(K^+\), we would have \(J^+\) simultaneously being the boundary of more than one domain, in addition to being the boundary of \(U^+={\mathbb C}^2-K^+\). This is not possible if \(J^+\) is a topological submanifold of \({\mathbb C}^2\).
By [FM] there are exactly d fixed points, counted with multiplicity. By Lemma 4.3, the fixed points in \(J^+\) are of saddle type, so they have multiplicity 1. Thus there are at least \(d-1\) of them. \(\square \)
5 Fixed Points with Given Multipliers
If \(q=(x,y)\) is a fixed point for \(f=f_n\circ \cdots \circ f_1\), then we may represent it as a finite sequence \((x_j,y_j)\) with \(j\in {\mathbb Z}/n{\mathbb Z}\), subject to the conditions \((x,y)=(x_1,y_1)=(x_{n+1},y_{n+1})\) and \(f_j(x_j,y_j)=(x_{j+1},y_{j+1})\). Given the form of \(f_j\), we have \(x_{j+1}=y_j\), so we may drop the \(x_j\)’s from our notation and write \(q=(y_n,y_1)\). We identify this point with the sequence \(\hat{q}=(y_1,\dots ,y_n)\in {\mathbb C}^n\), and we define the polynomials
The condition to be a fixed point is that \(\hat{q}=(y_1,\dots ,y_n)\) belongs to the zero locus \(Z(\varphi _1,\dots ,\varphi _n)\) of the \(\varphi _i\)’s. We define \(q_i(y_i):=p_i(y_i) - y_i^{d_i} \) and \(Q_i := q_i(y_i) - y_{i+1}-\delta _i y_{i-1}\), so
Since \(p_j\) is monic, the degrees of \(q_i\) and \(Q_i\) are \(\le d_i-1\).
By the Chain Rule, the differential of f at \(q=(y_n,y_1)\) is given by
We will denote this by \( M_n=M_n(y_1,\dots ,y_n):= \left( \begin{array}{cc}m_{11}^{(n)} &{} m_{12}^{(n)}\\ m_{21}^{(n)} &{} m_{22}^{(n)} \end{array}\right) \).
We consider special monomials in \(p'_j=p'_j(y_j)\) which have the form \((p')^L:= p'_{\ell _1}\cdots p'_{\ell _s}\), with \(L=\{\ell _1,\dots ,\ell _s\}\subset \{1,\dots ,n\}\). Note that the factors \(p'_{\ell _i}\) in \((p')^L\) are distinct. Let us use the notation |L| for the number of elements in L and \(H_\mathbf{m}\) for the linear span of \(\{(p')^L: |L|= m-2k, 0\le k\le {n/ 2} \}\). With this notation, m indicates the maximum number of factors of \(p'_j\) in any monomial, and in every case the number of factors differs from m by an even number.
Lemma 5.1
The entries of \(M_n\):
-
(1)
\(m_{11}^{(n)}\) and \(m_{22}^{(n)}- p'_1(y_1)\cdots p'_n(y_n)\) both belong to \( H_\mathbf{n-2}\).
-
(2)
\(m_{12}^{(n)}, m_{21}^{(n)} \in H_\mathbf{n-1}\).
Proof
We proceed by induction. The case \(n=1\) is clear. If \(n=2\),
which satisfies (1) and (2). For \(n>2\), we have
which gives (1) and (2) for all n. \(\square \)
The condition for Df to have a multiplier \(\lambda \) at q is \(\Phi (\hat{q})=0\), where
Lemma 5.2
\(\Phi -p'_1(y_1)\cdots p'_n(y_n) \in H_\mathbf{n-2}\).
Proof
The formula for the determinant gives
since \(\delta \) is the Jacobian determinant of Df. The Lemma now follows from Lemma 5.1. \(\square \)
The degree of the monomial \(y^a:= y_1^{a_1}\cdots y_n^{a_n}\) is \({\mathrm{deg}}(y^a)=a_1+\cdots + a_n\). We will use the graded lexicographical order on the monomials in \(\{y_1,\dots ,y_n\}\). That is, \(y^a> y^b\) if either \({\mathrm{deg}}(y^a)>{\mathrm{deg}}(y^b) \), or if \({\mathrm{deg}}(y^a)={\mathrm{deg}}(y^b) \) and \(a_i>b_i\), where \(i = \min \{1\le j\le n: a_j\ne b_j\}\). If \(f\in {\mathbb C}[y_1,\dots ,y_n]\), we denote LT(f) for the leading term of f, LC(f) for the leading coefficient, and LM(f) for the leading monomial.
Lemma 5.3
With the graded lexicographical order, \(G:=\{\varphi _1,\dots ,\varphi _n\}\) is a Gröbner basis.
Proof
We will use Buchberger’s Algorithm (see [10, Chapter 2]). For each \(i=1,\dots ,n\), LT\((\varphi _i)= \mathrm{LM}(\varphi _i)=y_i^{d_i}\), so for \(i\ne j\), the least common multiple of the leading terms is \(\mathrm{L.C.M.} = y_i^{d_i}y_j^{d_j}\). The S-polynomial is
where we use the \(Q_j\) from (4.1) and cancel terms. Now let \(\mu _i:={\mathrm{deg}}(Q_i)\). Since \(\mu _i<d_i\) for all i, the monomials LM\((\varphi _jQ_i)=y_j^{d_j}y_i^{\mu _i}\) and LM\((\varphi _iQ_j)=y_i^{d_i}y_j^{\mu _j}\) are not equal in our monomial ordering. Thus LM\((S(\varphi _i,\varphi _j)\ge \max (\mathrm{LM}(\varphi _j Q_i), \mathrm{LM}(\varphi _iQ_j))\). It follows from Buchberger’s Algorithm that \(\{\varphi _1,\dots ,\varphi _n\}\) is a Gröbner basis. \(\square \)
We will use the Multivariable Division Algorithm, by which any polynomial \(g\in {\mathbb C}[y_1,\dots ,y_n]\) may be written as \(g=A_1\varphi _1 + \cdots + A_n\varphi _n + R\) where LM\((g)\ge \mathrm{LM}(A_j\varphi _j)\) for all \(1\le j\le n\), and R contains no terms divisible by any LM\((\varphi _j)\). An important property of a Gröbner basis is that g belongs to the ideal \(\langle \varphi _1,\dots ,\varphi _n\rangle \) if and only if \(R=0\) (see, for instance, [10] or [1]).
If all fixed points have the same value of \(\lambda \) as multiplier, then it follows that \(\Phi \) must vanish on the whole zero set \(Z(\varphi _1,\dots ,\varphi _n)\). Since we have a Gröbner basis, we easily determine the following:
Corollary 5.4
\(\Phi \notin \langle \varphi _1,\dots ,\varphi _n\rangle \).
Proof
The leading monomial of \(\Phi \) is \(y_1^{d_1-1}\cdots y_n^{d_n-1}\), but this is not divisible by any of the leading monomials LM\((\varphi _j)=y_j^{d_j}\). Since \(\{\varphi _1,\dots ,\varphi _n\}\) is a Gröbner basis, it follows that \(\Phi \) does not belong to the ideal \(\langle \varphi _1,\dots ,\varphi _n\rangle \). \(\square \)
6 Proof of the Theorem
In this section we prove the Theorem, which will follow from 4.3, in combination with:
Proposition 6.1
Suppose \(F=f_n\circ \cdots \circ f_1\), \(n\ge 3\), is a composition of generalized Hénon maps with \(|\delta |<1\). Suppose that F has \(d=d_1\cdots d_n\) distinct fixed points. It is not possible that \(d-1\) of these points have the same multipliers.
Proof that Proposition 6.1 implies the Theorem
To prove the Theorem, it remains to deal with the case \(|\delta |<1\). If \(f=f_1\) is a single generalized Hénon map, we consider \(F=f_1\circ f_1\circ f_1\) with \(n=3\) and the same Julia set. Lemma 3.4 asserts that if \(J^+\) is \(C^1\), there are \(d-1\) saddle points with unstable multiplier \(\lambda =d\). So by Proposition 4.1, we conclude that \(J^+\) cannot be \(C^1\) smooth. \(\square \)
We give the proof of Proposition 6.1 at the end of this section. For \(J\subset \{1,\dots ,n\}\), we write
We let \(H_J\) denote the linear span of \(\Lambda _J\). To compare with our earlier notation, we note that \(H_J\subset H_\mathbf{|J|-2}\) and that \((p')^J\notin H_J\). The elements of \(H_J\) depend only on the variables \(y_j\) for \(j\in J\). Now we formulate a result for dividing certain terms by \(\varphi _j\):
Lemma 6.2
Suppose that \(J\subset \{1,\dots ,n\}\) and \(h\in H_J\). Then for each \(j\in J\) and \(\alpha \in {\mathbb C}\), we have
where \(\rho _1,\rho _2\in H_{J-\{j\}}\), and \(B=\eta _j(y_j) + d_j y_{j+1} + d_j\delta _j y_{j-1} \) with
The leading monomials satisfy
Proof
Let us start with the case \(J=\{1,\dots ,m\}\), \(m\le n\), and \(j=1\), so \(J-\{j\}=J_{\hat{1}}=\{2,\dots ,n\}\). We divide by \(p'_1\) and remove any factor of \(p'_1\) in h. This gives
where \(\mu _1=(p')^{J_{\hat{1}}} +\rho _1\), \(\rho _1,\rho _2\in H_{\{2,\dots ,m\}}\), and \(\mu _1\), \(\rho _1\), \(\rho _2\) are independent of the variable \(y_1\). Thus
where in the last line we substitute \(\eta _1\) defined by \((\ddag )\). Using \((*)\), we see that this gives \((\dag )\).
It remains to look at the leading terms of \(T_1:= (y_1-\alpha )\left( (p')^J + h) \right) \) and \(T_2:= d_1\mu _1\varphi _1\). We see that \(T_1\) and \(T_2\) both contain nonzero multiples of \(y_j \prod _{i=1}^m y_i^{d_i -1}\), and all other monomials in \(T_1\) and \(T_2\) have lower degree. Thus we have \(LM(T_1)=LM(T_2)\) for the graded ordering, independent of any ordering on the variables \(y_1,\dots ,y_n\). The choices of \(J=\{1,\dots ,m\}\) and \(j=1\) just correspond to a permutation of variables, and this does not affect the conclusion that \(LM(T_1)=LM(T_2)\). \(\square \)
Lemma 6.3
For any \(\alpha \in {\mathbb C}\), \((y_1-\alpha )\Phi \notin \langle \varphi _1,\dots ,\varphi _n\rangle \).
Proof
By [FM], we may assume that \(p_j(y_j)=y_j^{d_j} + q_j(y_j) \) and \({\mathrm{deg}}(q_j)\le d_j-2\). We consider two cases. The first case is that there is at least one j such that \(\eta _j\) is not the zero polynomial. If we conjugate by \(f_{j-1}\circ \cdots \circ f_1\), we may “rotate” the maps in f so that the factor \(f_j\) becomes the first factor. If there exists a j for which \(\eta _j(y_j)\) is nonconstant, we choose this for \(f_1\). Otherwise, if all the \(\eta _j\) are constant, we choose \(f_1\) to be any factor such that \(\eta _1\ne 0\).
We will apply the Multivariate Division Algorithm on \((y_1-\alpha )\Phi \) with respect to the set \(\{\varphi _1,\dots ,\varphi _n\}\). We will find that there is a nonzero remainder, and since \(\{\varphi _1,\dots ,\varphi _n\}\) is a Gröbner basis, it will follow that \((y_1-\alpha )\Phi \) does not belong to the ideal \(\langle \varphi _1,\dots ,\varphi _n\rangle \).
We start with Lemma 5.2, according to which \(\Phi = p_1'\cdots p_n' + h\), where \(h\in H_\mathbf{n-2} = H_{\{1,\dots ,n\}}\). The leading monomial of \((y_1-\alpha )\Phi \) is \(y_1^{d_1}\prod _{i=2}^n y_i^{d_i-1}\), and \(\varphi _1\) is the only element of the basis whose leading monomial divides this. Thus we apply Lemma 6.2, with \(J=\{1,\dots ,n\}\), \(j=1\), and \(J_{\hat{1}}:= J-\{j\}=\{2,\dots ,n\}\). This gives
where \(\rho _1,\rho _2\in H_{\{2,\dots ,n\}}\). In particular, \(T_2\) and \(T_n\) depend on \(y_2,\dots ,y_n\) but not on \(y_1\). We note that \(T_2\) (respectively, \(T_n\)) contains a term divisible by LM\((\varphi _2)\) (respectively, LM\((\varphi _n)\)). We view \(R_1\) as a remainder term, and note that LM\((R_1)\) is divisible by \(y_2^{d_2-1}\cdots y_n^{d_n-1}\), as well as the largest power of \(y_1\) in \(\eta _1(y_1)\). By “\(\ell .o.t.\),” we mean that none of its monomials is divisible by LM\((R_1)\) or by any of the LM\((\varphi _j)\).
Now we apply Lemma 6.2 to \(T_2\), this time with \(J=\{2,\dots ,n\}\) and \(j=2\), with \(J-\{2\} = J_{\hat{1}\hat{2}} = \{3,\dots ,n\}\). We have
We see that \(T_2^{(2)}\) contains terms that are divisible by LM\((\varphi _3)\), but the monomials in \(R_1^{(2)}\) and \(R_2^{(2)}\) are not divisible by LM\((\varphi _i)\) for any i. The remainder term here is \(R_1^{(2)}+R_2^{(2)}\), and we observe that this cannot cancel the largest term in \(R_1\). This is because LM\((R_1^{(2)})\) lacks a factor of \(y_2\), and LM\((R_2^{(2)})\) is equal to \(y_3^{d_3-1}\cdots y_n^{d_n-1}\) times the largest power of \(y_2\) in \(\eta _2(y_2)\), and by \((\ddag )\), this power is no bigger than \(d_2-1\). If \(\eta _1\) is not constant, then we see that LM\((R_1)>\mathrm{LM}(R_2^{(2)})\). If \(\eta _1\) is constant, then \(\eta _2\) must be constant, too, and again we have LM\((R_1)>\mathrm{LM}(R_2^{(2)})\). Thus, with our earlier notation, \(R_1^{(2)} + R_2^{(2)} = \ell .o.t.\)
We do a similar procedure with \(T_n\), \(T_2^{(2)}\), etc., and again find that the remainder term does not contain a multiple of the leading monomial of \(R_1\). We see that each time we do this process, the size of the exponent L decreases in the term \((p')^L\). When we have \(L=\emptyset \), there are no terms that can be divided by any LM\((\varphi _j)\). Thus we end up with
and LT\(((y_1-\alpha )\Phi )\ge \mathrm{LT}(A_j\varphi _j)\) for all \(1\le j\le n\), and none of the remaining terms is divisible by any of the leading monomials of \(\varphi _j\). Thus we have now finished the Multivariate Division Algorithm, and we have a nonzero remainder. Thus \((y_1-\alpha )\Phi \) does not belong to the ideal of the \(\varphi _j\)’s.
Now we turn to the second case, in which \(\eta _j=0\) for all j. By [12], we may assume that \({\mathrm{deg}}(q_j)\le d_j-2\). It follows that \(\alpha =0\) and \(q_j=0\). Thus \(p_j = y_j^{d_j}\) for all \(1\le j\le n\), so \(p_j' = d_j y_j^{d_j-1}\), and \(H_J\) consists of linear combinations of products \((p')^I= y_{i_1}^{d_{i_1}-1}\cdots y_{i_k}^{d_{i_k}-1}\) for \(I = \{i_1,\dots ,i_k\}\subset J\), for even \(k\le |J|-2\). We will go through the Multivariate Division Algorithm again. The principle is the same as before, but the details are different; in the first case we needed \(n\ge 2\), and now we will need \(n\ge 3\).
Again, it is only \(\varphi _1\) which has a leading monomial which can divide some terms in \((y_1-\alpha )\Phi \). As before, we apply Lemma 6.2 with \(J=\{1,\dots ,n\}\), \(j=1\), and \(J-\{1\}=J_{\hat{1}}=\{2,\dots ,n\}\). The polynomial in \((\ddag )\) becomes \(B=d_jy_{j+1} + d_j\delta _j y_{j-1}\), and we have
where \(\rho _1,\rho _2\in H_{\{2,\dots ,n\}}\). Now we apply Lemma 6.2 to divide \(T_2\) (respectively, \(T_n\)) by \(\varphi _2\) (respectively, \(\varphi _n\)). This yields
where
with \(\tilde{\rho }_3\in H_{\{3,\dots ,n\}}\) and \(\tilde{\rho }_n\in H_{\{2,\dots ,n-1\}}\), and
Since \(n>2\), R is not the zero polynomial. We will continue the Multivariate Division Algorithm by dividing \(T_3\) by \(\varphi _3\) and \(T_n\) by \(\varphi _n\), but we see that any terms created cannot cancel R. Thus when we finish the Multivariable Division Algorithm, we will have a nonzero remainder. As in the previous case, we conclude that \(y_1\Phi \) is not in the ideal \(\langle \varphi _1,\dots ,\varphi _n\rangle \).
Proof of Proposition 6.1
The fixed points of f coincide with the elements of \(Z(\varphi _1,\dots ,\varphi _n)\), which is a variety of pure dimension zero. Saddle points have multiplicity 1, and since there are \(d-1\) of these, and since the total multiplicity is d, there must be one more fixed point, also of multiplicity 1. It follows that the ideal \(I:=\langle \varphi _1,\dots ,\varphi _n\rangle \) is equal to its radical (see [1]). Since the saddle points all have multiplier \(\lambda \), \(\Phi \) must vanish at all the saddle points. If \((\alpha ,\beta )\) is the other fixed point, we conclude that \((y_1-\alpha )\Phi \) vanishes at all the fixed points. Thus \((y_1-\alpha )\Phi \) belongs to the radical of I, and thus I itself. This contradicts Lemma 6.3, which completes the proof of Proposition 6.1. \(\square \)
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Acknowledgements
We wish to thank Yutaka Ishii and Paolo Aluffi for helpful conversations on this material.
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Appendix: Nonsmoothness of J, \(J^*\), and K
Appendix: Nonsmoothness of J, \(J^*\), and K
Let us turn our attention to other dynamical sets for polynomial diffeomorphisms of positive entropy. These are \(J:=J^+\cap J^-\), \(K:= K^+\cap K^-\), and the set \(J^*\), which coincides with the closure of the set of periodic points of saddle type. (See [3, 5], and [2] for other characterizations of \(J^*\).) We have \(J^*\subset J\subset K\). We note that none of these sets can be a smooth 3-manifold: otherwise, for any saddle point p, it would be a bounded set containing \(W^s(p)\) or \(W^u(p)\), which is the holomorphic image of \({\mathbb C}\). The following was suggested by Remark 5.9 of Cantat in [9]; we sketch his proof:
Proposition 6.1
If \(J=J^*\), then it is not a smooth 2-manifold.
Proof
Let p be a saddle point, and let \(W^u(p)\) be the unstable manifold. The slice \(J \cap W^u(p)\) is smooth and invariant under multiplication by the multiplier of Df. This means that in fact, the multiplier must be real, and the restriction of \(G^+\) to the slice must be linear on each (half-space) component of \(W^u(p)- J\).
The identity \(G^+\circ f = d\cdot G^+\) means that the canonical metric (defined in [6]) is multiplied by d. Thus f is quasi-expanding on \(J^*\). Now, applying this argument to \(f^{-1}\) we get that f is quasi-hyperbolic. Further, \(J^*=J\), so it is quasi-hyperbolic on J. If f fails to be hyperbolic, then by [7] there will be a one-sided saddle point, which cannot happen since J is smooth.
Now that f is hyperbolic on J, there is a splitting \(E^s \oplus E^u\) of the tangent bundle, so we conclude that J is a 2-torus. The dynamical degree must be the spectral radius of an invertible 2-by-2 integer matrix, but this means it is not an integer, which contradicts the fact the dynamical degree of a Hénon map is its algebraic degree.
Proposition 6.2
Suppose that the complex Jacobian is not equal to \(\pm 1\). Then for each saddle (periodic) point p and each neighborhood U of p, neither \(J\cap U\) nor \(J^*\cap U\) nor \(K\cap U\) is a \(C^1\) smooth 2-manifold.
Proof
Let us write \(M:=J\cap U\) and \(g:=f|_M\). (The following argument works, too, if we take \(M=J^*\cap U\) or \(M=K\cap U\).) The tangent space \(T_pM\) is invariant under Df. The stable/unstable spaces \(E^{s/u}\subset T_p{\mathbb C}^2\) are invariant under \(D_pf\). The space \(E^s\) (or \(E^u\)) cannot coincide with \(T_pM\), for otherwise the complex stable manifold \(W^s(p)\) (or \(W^u(p)\)) would be locally contained in M, and thus globally contained in J. But the \(W^{s/u}\) are uniformized by \({\mathbb C}\), whereas J is bounded. We conclude that p is a saddle point for g, and thus the local stable manifold \(W^s_{\mathrm{loc}}(p; g)\) is a \(C^1\)-curve inside the complex stable manifold \(W^s(p)\). As in Lemma 4.3, we conclude that the multiplier for \(D_pf|_{E^u_p}\) is \(\pm d\) and the multiplier for \(D_pf|_{E^s_p}\) is \(\pm 1/d\). Thus the complex Jacobian is \(\delta =\pm 1\).
Solenoids The two results above concern smoothness, but no example is known where J, \(J^*\), or K is even a topological 2-manifold. In the cases where \(J^+\) has been shown to be a topological 3-manifold (see [8, 11, 16, 20]) it also happens that J is a (topological) real solenoid, and in these cases it is also the case that \(J=J^*\). Further, for every saddle (periodic) point p, there is a real arc \(\gamma _p=W^u_{\mathrm{loc}}(p)\cap J\). If we apply the argument of Proposition 6.2 to this case, we conclude that \(\gamma _p\) is not \(C^1\) smooth.
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Bedford, E., Kim, K. No Smooth Julia Sets for Polynomial Diffeomorphisms of \({\mathbb C}^2\) with Positive Entropy. J Geom Anal 27, 3085–3098 (2017). https://doi.org/10.1007/s12220-017-9796-1
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DOI: https://doi.org/10.1007/s12220-017-9796-1