On diffeomorphisms of compact 2-manifolds with all nonwandering points periodic

TL;DR

This paper investigates conditions under which all non-wandering points in a 2D dynamical system are periodic, including a survey, a negative counterexample, and a positive condition for the Hénon family.

Contribution

It provides a survey of known results, constructs a counterexample diffeomorphism, and establishes a condition ensuring all non-wandering points are periodic in the Hénon family.

Findings

Constructed a Kupka--Smale diffeomorphism with nonwandering points not all periodic

Identified a condition on the Hénon family guaranteeing all nonwandering points are periodic

Discussed open problems and future directions in the study of 2D dynamical systems

Abstract

The aim of the present paper is to study conditions under which all the non-wandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one positive and one negative. The negative result: we construct a Kupka--Smale diffeomorphism in $\mathbb{R}^2$ (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied H\'{e}non family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.