Strongly dissipative surface diffeomorphisms

TL;DR

This paper introduces a new class of volume-contracting surface diffeomorphisms with intermediate dynamics, providing a reduced model and a closing lemma, including Hénon maps with small Jacobian.

Contribution

It defines a novel class of dissipative surface diffeomorphisms and proves a $C^ abla$-closing lemma, linking dynamics to a one-dimensional model, and includes Hénon maps with small Jacobian.

Findings

Established a reduced one-dimensional model for the class

Proved a $C^ abla$-closing lemma for ergodic measures

Included Hénon maps with Jacobian in $(-1/4,1/4)$

Abstract

We introduce a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems one can associate to the dynamics a reduced one-dimensional model and it is proved a type of $C^\infty-$closing lemma on the support of every ergodic measure. We also show that this class contains H\'enon maps with Jacobian in $(-1/4,1/4)$.