A rigidity theorem for H\'{e}non maps

TL;DR

This paper investigates the properties of Hénon maps sharing certain non-escaping sets and explores the complex structure of specific domains, revealing new rigidity phenomena and examples of inequivalent complex domains.

Contribution

It establishes a rigidity theorem for Hénon maps with shared non-escaping sets and constructs a continuum of inequivalent Short-ℂ² domains, including non-Reinhardt examples.

Findings

Shared non-escaping sets imply rigidity in Hénon maps

Existence of a continuum of inequivalent Short-ℂ² domains

Examples of non-Reinhardt Short-ℂ² domains

Abstract

The purpose of this note is two fold. First, we study the relation between a pair of H\'{e}non maps that share the same forward and backward non-escaping sets. Second, it is shown that there exists a continuum of $Short-\mathbb{C}^2$'s that are biholomorphically inequivalent and finally, we provide examples of $Short-\mathbb{C}^2$'s that are neither Reinhardt nor biholomorphic to Reinhardt domains.