Relation between H\'{e}non maps with biholomorphic escaping sets

Abstract

We study H\'{e}non maps with biholomorphic escaping sets. We show that if $H$ and $F$ are two H\'{e}non maps of degree $d$ with biholomorphic escaping sets, then there exist complex numbers $\alpha,\beta$ and $\gamma$ with $\alpha^{d+1}=\beta$ and $\beta^{d-1}=\gamma^{d-1}=1$ such that with the following linear automorphisms $$B(x,y)=(\gamma \alpha \beta^{-1}x, \alpha^{-1}y) \quad\text{and}\quad L(x,y)=(\gamma^{-1}\beta x,\beta y )$$one has $$ F\equiv L \circ B \circ H \circ B. $