Dynamics of transcendental H\'enon maps III: Infinite entropy

TL;DR

This paper investigates transcendental Hénon maps in several complex variables, proving they have infinite entropy and infinitely many periodic orbits, thus advancing understanding of their complex dynamics.

Contribution

It establishes that transcendental Hénon maps possess infinite topological and measure-theoretic entropy, and confirms the existence of infinitely many periodic orbits of any order greater than two.

Findings

Transcendental Hénon maps have infinite entropy.

Existence of infinitely many periodic orbits of any order > 2.

Bridges ideas from one-variable transcendental dynamics and polynomial Hénon maps.

Abstract

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental H\'enon maps offers the potential of combining ideas from transcendental dynamics in one variable, and the dynamics of polynomial H\'enon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.