Aperiodic chain recurrence classes of $C^1$-generic diffeomorphisms

TL;DR

This paper investigates the complex topological behaviors of aperiodic chain recurrence classes in $C^1$-generic diffeomorphisms on three-dimensional manifolds, revealing their diverse dynamical properties.

Contribution

It demonstrates that $C^1$-generic diffeomorphisms can have aperiodic classes with various intricate topological and ergodic properties, expanding understanding of dynamical complexity.

Findings

Existence of minimal expansive aperiodic classes

Presence of minimal but non-uniquely ergodic aperiodic classes

Occurrence of transitive but non-minimal aperiodic classes

Abstract

We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are $C^1$-generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes.