On the group of automorphisms of the escaping sets of Hénon maps

Abstract

In this article, we provide an explicit description of ${\mathrm{Aut}}_1(I_H^{+})$, the group of automorphisms of the escaping set $I_H^{+}$ of a Hénon map that induce the identity on its fundamental group. More precisely, for a Hénon map $H(x,y)=(y,p(y)-\delta x),$ where $p$ is a polynomial of degree $d\ge 2$ and $\delta \ne 0$, we show that ${\mathrm{Aut}}_1(I_H^{+})$ is completely determined by the degree and the coefficients of $p$. Thus, the automorphism group of the escaping set, subject to the prescribed topological constraint, can be explicitly characterized in terms of the underlying Hénon map.