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Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$

Sayani Bera, Kaushal Verma

TL;DR

The paper proves a rigidity phenomenon for the escaping set $U^+$ of a generalised Hénon map $H$ with positive entropy: any $f\in\mathrm{Aut}(\mathbb{C}^2)$ preserving $U^+$ must be of the form $f=L\circ H^s$ with an affine $L$ preserving $K^{\pm}$ and $s\in\mathbb{Z}$. The authors construct an intermediate cover $\widehat{U}^+$, biholomorphic to $\mathbb{C}\times(\mathbb{C}\setminus\overline{\mathbb{D}})$, and analyze the lifted dynamics $\widehat{H}$ and deck transformations; this reduces the automorphism problem to a finite cyclic structure controlled by the degrees $d$ and $d'$. They show Aut$_1(U^+)$ is a semidirect product $\mathbb{C}\rtimes G$ with $G\le\mathbb{Z}_{d_0(d-1)}$, and that Aut$(U^+)\cap$Aut$(\mathbb{C}^2)$ is finite cyclic, with the same finite group governing automorphisms of punctured Short $\mathbb{C}^2$ sublevel sets $\Omega_c'$. Consequently, all holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set, and the automorphism groups are independent of $c>0$. The paper also establishes an equivalence between automorphisms of $U^+$ and those of the punctured Short domains, yielding rigidity results: biholomorphic equivalence of $\Omega_c'$ implies affine conjugacy of the underlying Hénon maps, and any $\phi\in\mathrm{Aut}(\mathbb{C}^2)$ preserving $K^+$ is constrained to be affine-conjugate to a power $H^s$ of the map.

Abstract

Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets $\{G^+ < c\}$, $c > 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group.

Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$

TL;DR

The paper proves a rigidity phenomenon for the escaping set of a generalised Hénon map with positive entropy: any preserving must be of the form with an affine preserving and . The authors construct an intermediate cover , biholomorphic to , and analyze the lifted dynamics and deck transformations; this reduces the automorphism problem to a finite cyclic structure controlled by the degrees and . They show Aut is a semidirect product with , and that AutAut is finite cyclic, with the same finite group governing automorphisms of punctured Short sublevel sets . Consequently, all holomorphic automorphisms of these Short s are affine automorphisms of preserving the escaping set, and the automorphism groups are independent of . The paper also establishes an equivalence between automorphisms of and those of the punctured Short domains, yielding rigidity results: biholomorphic equivalence of implies affine conjugacy of the underlying Hénon maps, and any preserving is constrained to be affine-conjugate to a power of the map.

Abstract

Let be a polynomial automorphism of of positive entropy and degree . We prove that the escaping set (or equivalently, the non-escaping set ), of is rigid under the action of holomorphic automorphisms of . Specifically, every holomorphic automorphism of that preserves takes the form where and belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets , , of the Greens function associated with the map are canonical examples of Short s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short s are affine automorphisms of preserving the escaping set . Hence, the automorphism group of these Short s are the same for every and is a finite cyclic group.
Paper Structure (14 sections, 26 theorems, 226 equations, 1 figure)

This paper contains 14 sections, 26 theorems, 226 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An intermediate cover of the escaping set U+
  3. Construction of the cover
  4. The form of the map on an appropriate subdomain
  5. is biholomorphic to
  6. Deck transformations of
  7. Aut(U+) and Aut(Om-c)
  8. The subgroup Aut1(U+)
  9. The group
  10. The automorphism groups of U+ and Om-c
  11. The subgroup and Proof of Theorem \ref{['t:main']}
  12. The group
  13. The group
  14. Proof of Theorem \ref{['t:equivalence of automorphisms']} and Corollary \ref{['c:automorphism of C2']}

Key Result

Theorem 1.1

Let $H$ be a generalised Hénon map of the form (e:henon), i.e., where $H_i(x,y)=(y, p_i(y)-a_i x)$, $p_i$ is a polynomial with degree $d_i \ge 2$ and $a_i \neq 0$ for every $1 \le i \le m$. Then the non-escaping set $K^+$, or equivalently, the escaping set $U^+$, associated with $H$ is rigid under the action of ${\rm Aut}(\mathbb{C}^2)$. In other words, any $f \ is of the form $f=L \circ H^s\text

Figures (1)

  • Figure 1: Behaviour of $\widetilde{\Psi}$ on $\widetilde{W}_M^+$ and the image of $\widetilde{\Psi}(0,y)$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 41 more