Stability and bifurcations for dissipative polynomial automorphisms of ${{\mathbb {C}}^2}$

Abstract

We study stability and bifurcations in holomorphic families of polynomial automorphisms of ${{\mathbb {C}}^2}$. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of $J$-stability in one-dimensional dynamics. Define the bifurcation locus to be the complement of the weak stability locus. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis’ Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of “critical points” in semi-parabolic basins (which makes use of the classical Denjoy–Carleman–Ahlfors and Wiman Theorems).

Appendix A. Attracting basins

The methods of Sect. 7.2 also give the existence of “critical points” in attracting basins, under certain minor hypotheses (that are needed to even define the critical points). Though these results are not used in the paper, they are interesting on their own right.

Let $f$ be a polynomial automorphism of dynamical degree $d\ge 2$ with an attracting point $p$. As usual, we may assume that $p$ is fixed, and we denote by $\mathcal {B}$ its basin of attraction. It is classical that there is a local holomorphic change of coordinates which puts $f$ in a simple normal form (this result goes apparently back to Lattès [36]). Let $\kappa _1$ and $\kappa _2$ be the eigenvalues of $Df_p$, ordered so that $0< \left| \kappa _2\right| \le \left| \kappa _1\right| <1$. We say that $(\kappa _1, \kappa _2)$ is resonant if there exists an integer $i\ge 1$ such that $\kappa _2 = \kappa _1^i$ (notice that $i=1$ is allowed). Then there exists a local change of coordinates near $p$ such that in the new coordinates $(z_1, z_2)$, $f$ expresses as

$$\begin{aligned} f(z_1, z_2 ) = {\left\{ \begin{array}{ll} (\kappa _1 z_1, \kappa _2 z_2) \text { if }(\kappa _1, \kappa _2) \text { is not resonant,}\\ (\kappa _1 z_1, \kappa _2 z_2+ \alpha z_1^i) \text { otherwise, where }i\text { is as above,}\\ \quad \text { and }\alpha \in \{0,1\}. \end{array}\right. } \end{aligned}$$

In any case, we see that the vertical foliation $\{z_1 = C\}$ is invariant under $f$. If $\left| \kappa _2\right| <\left| \kappa _1\right|$ this is the “strong stable foliation”, characterized by the property that points in the same leaf approach each other at the fastest possible rate $\kappa _2^n$. As before, it will be denoted by $\mathcal {F}^{ss}$. Using the dynamics, the coordinates $(z_1, z_2)$ extend to the basin and define a biholomorphism $\mathcal {B}\simeq {{\mathbb {C}}^2}$. In the non-resonant (i.e. linearizable) case, the foliation $\{z_2 = C\}$ is invariant as well. We then simply refer to $\{z_1=C\}$ and $\{z_2=C\}$ as the invariant coordinate foliations in $\mathcal {B}$.

We give two statements on the existence of critical points. The first one parallels Theorem B

Theorem A.1

Let $f$ be a polynomial automorphism of ${{\mathbb {C}}^2}$ of dynamical degree $d\ge 2$, possessing an attracting point $p$, whose eigenvalues satisfy $0<\left| \kappa _2\right| <\left| \kappa _1\right| <1$, with basin of attraction $\mathcal {B}$. Assume that $\left| \mathrm{Jac }\,f\right| <{d^{-4}}$, or more generally that the connected component of $p$ in $W^{ss}(p)\cap J^-$ is $\{p\}$. Then for every saddle periodic point $q$, every component of $W^u(q)\cap \mathcal {B}$ contains a critical point, that is, a point of tangency with the strong stable foliation in $\mathcal {B}$.

The second statement concerns the hyperbolic case.

Theorem A.2

Let $f$ be a polynomial automorphism of ${{\mathbb {C}}^2}$ of dynamical degree $d\ge 2$, possessing an attracting point $p$ with basin $\mathcal {B}$. Assume that $f$ is uniformly hyperbolic on $J$, and fix any saddle periodic point $q$.

If the eigenvalues of $p$ satisfy $\left| \kappa _2\right| <\left| \kappa _1\right|$ (resp. are non-resonant), then every component of $W^u(q)\cap \mathcal {B}$ admits a tangency with the strong stable foliation of $\mathcal {B}$ (resp. with both invariant coordinate foliations).

Here is an interesting geometric consequence. Recall that if $f$ is dissipative and hyperbolic, $J^+$ is (uniquely) laminated by stable manifolds. Let us denote by $W^s(J)$ this lamination. It is natural to wonder whether the strong stable foliation in $\mathcal {B}$ matches continuously with the lamination of $J^+$ (recall that $\partial \mathcal {B} =J^+$). The existence of critical points implies that this is never the case (compare [4, Cor. A.2]).

Corollary A.3

Let $f$ be as in the previous theorem, in particular $f$ is hyperbolic on $J$. Then if $p$ is an attracting point with eigenvalues $\left| \kappa _2\right| <\left| \kappa _1\right|$ and basin $\mathcal {B}$, then for every $x\in J$, $W^s(J)\cup \mathcal {F}^{ss}(\mathcal {B})$ does not define a lamination near $x$. If $p$ is linearizable, the same holds for both invariant coordinate foliations.

Proof

Let us deal with the case where $\left| \kappa _1\right| <\left| \kappa _2\right|$. It is enough to assume that $x$ is a saddle periodic point. Hyperbolicity implies that $W^u(J)$ and $W^s(J)$ are transverse near $x$, so if $W^s(J)\cup \mathcal {F}^{ss}(\mathcal {B})$ is a lamination near $x$, $\mathcal {F}^{ss}(\mathcal {B})$ must be transverse to $W^u(J)$ near $x$. On the other hand, there exist critical points on $W^u(x)$ arbitrary close to $x$ (obtained from the previous ones by iterating backwards). This contradiction finishes the proof. $\square$

Proof of Theorems A.1 and A.2

This is very similar to Proposition 7.8 so the proof is merely sketched. Let us first deal with the case where $\left| \kappa _2\right| <\left| \kappa _1\right|$, with $f$ hyperbolic or not. Let $\pi _1:\mathcal {B}\rightarrow {\mathbb {C}}$ be the projection along the strong stable foliation. In the coordinates $(z_1, z_2)$, it simply expresses as $(z_1, z_2)\mapsto z_1$. Assume by contradiction that there is no critical point in $\Omega$. Then $\pi _1\circ \psi ^u: \Omega {\setminus } (\psi ^u)^{-1} (W^{ss}(p)) \rightarrow {\mathbb {C}}^*$ is a locally univalent map. Since it cannot be a covering it must possess an asymptotic value, hence there is a diverging path $\gamma$ in $\Omega$ such that the limit $\lim _{t\rightarrow \infty } \pi _1\circ \psi ^u(\gamma (t)) = \omega$ exists in ${\mathbb {C}}^*$. Let $\pi _2:\mathcal {B}\rightarrow {\mathbb {C}}$ be the second coordinate projection. As before, we split the argument according to the bounded or unbounded character of $\pi _2\circ \psi ^u(\gamma )$.

If $\pi _2\circ \psi ^u(\gamma )$ is unbounded, we iterate forward and take cluster values to create an unbounded component $C$ of $J^-\cap W^{ss}(p)$ containing $p$. Now if $\left| \mathrm{Jac }\,f\right| < d^{-4}$, then $\left| \kappa _2\right| <d^{-2}$, so by Corollary 7.7, the component of $p$ in $W^{ss}(p)\cap J^-$ is a point, and we get a contradiction.

If $f$ is hyperbolic we argue as follows: in $\mathcal {B}{\setminus }\{p\}$, $J^-$ is laminated by unstable manifolds. In particular by [7, Lemma 6.4] the set of tangencies between $W^{ss}(p)$ and the unstable lamination is discrete. Pick $c\in C{\setminus }\{p\}$ such that $W^{ss}(p)$ and the unstable lamination are transverse near $c$. There exist coordinates $(x,y)$ close to $c$ in which $W^{ss}(p)$ is $\{x=0\}$, $c=(0,0)$, and the leaves of the unstable lamination close to $c$ are horizontal in the unit bidisk $\mathbb {B}$. By construction, there is a sequence of integers $n_j$ such that $f^{n_j}(\psi ^u(\gamma ))$ has a component $C_j$, vertically contained in $\mathbb {B}$, touching the boundary, and passing close to $c$. On the other hand $C_j$ must be contained in a leaf of the unstable foliation so we get a contradiction.

If $\pi _2\circ \psi ^u(\gamma )$ is bounded, then as before the path $\gamma$ must be unbounded in $W^u(q)$. Let $E$ be the cluster set of $\psi ^u(\gamma )$, which is a compact subset of the strong stable leaf $\{z_1 =\omega \}$. If $\left| \kappa _2\right| <\left| \kappa _1\right|$, then as in Proposition 7.8 we make a linear change of coordinates close to $p$ such that in the new coordinates, $f$ expresses as $f(x,y ) = (\kappa _1 x, \kappa _2y)+ h.o.t.$ We see that in these coordinates, $\mathrm {pr_1}(f^n(E))$ is a set of diameter $\lesssim \kappa _2^n$ about $x_n\sim c \kappa _1^n$, which leads to a contradiction with Theorem 7.1, exactly as in Proposition 7.8.

It remains to treat the case where $f$ is hyperbolic, $p$ is linearizable, and we look for tangencies with any of the invariant coordinate foliations. We argue exactly as before, with $(\pi _1, \pi _2)$ being the linearizing coordinate projections, in any order, and keep the same notation. The case where $\pi _2\circ \psi ^u(\gamma )$ is unbounded is dealt with exactly as above. If now $\pi _2\circ \psi ^u(\gamma )$ is bounded and $E$ denotes its cluster set in the leaf $\{z_1 = \omega \}$, we observe that as in the unbounded case, the laminar structure of $J^-$ outside $p$ forces $E$ to be reduced to a point. Therefore $\psi ^u$ admits an asymptotic value in $\mathcal {B} {\setminus }\{p\}$ and the contradiction arises by iterating and applying the ordinary Denjoy–Carleman–Ahlfors theorem.   $\square$