Hyperbolic sets that are not contained in a locally maximal one

TL;DR

This paper constructs new examples of hyperbolic sets that are not contained in any locally maximal hyperbolic set, answering longstanding questions and demonstrating robustness in three-dimensional tori.

Contribution

It provides the first transitive, robust example in three dimensions and shows such examples cannot exist in two dimensions.

Findings

Existence of hyperbolic sets not contained in any locally maximal set in $ ext{T}^3$

Robustness of these examples in linear Anosov diffeomorphisms

Non-existence of such examples in dimension 2

Abstract

In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt. Let $\Lambda$ be a hyperbolic set, and let $V$ be an open neighborhood of $\Lambda$. Does there exist a locally maximal hyperbolic set $\widetilde{\Lambda}$ such that $\Lambda \subset \widetilde{\Lambda} \subset V $? We show that such examples are present in linear anosov diffeomorophisms of $\mathbb{T}^3$, and are therefore robust. Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier and by Fisher, and these were either in dimension bigger than 4 or they were not transitive. We give a transitive and robust example in $\mathbb{T}^3$. And show that such examples cannot be build in dimension 2.