A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

TL;DR

This paper introduces a $C^1$-perturbation method to remove specific periodic points from a chain recurrence class in partially hyperbolic systems, using Markov iterated function systems as an intermediate step.

Contribution

It develops a novel perturbation technique for selectively ejecting periodic points while preserving certain intersections in partially hyperbolic dynamics.

Findings

Technique successfully removes targeted periodic points.

Applicable to three-dimensional partially hyperbolic diffeomorphisms.

Provides a new tool for controlling chain recurrence classes.

Abstract

We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/hetero-clinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called Markov iterated function system, a two dimensional iterated function system in which the compositions are chosen in Markovian way. Then we apply the result to the setting of three dimensional partially hyperbolic diffeomorphisms.