Hyperbolic Sets and Entropy at the Homological Level

TL;DR

This paper investigates the relationship between hyperbolic sets and entropy in partially hyperbolic diffeomorphisms on tori, proposing a refined entropy conjecture and exploring algebraic and dynamical indices.

Contribution

It establishes a connection between hyperbolic set indices and algebraic indices, and provides examples to guide higher-dimensional center case studies.

Findings

Link between unstable hyperbolic set index and algebraic index.

Examples illustrating key questions in higher-dimensional center cases.

Abstract

The aim of this work is to study a kind of refinement of the entropy conjecture, in the context of partially hyperbolic diffeomorphisms with one dimensional central direction, of d-dimensional torus. We start by establishing a connection between the unstable index of hyperbolic sets and the index at algebraic level. Two examples are given which might shed light on which are the good questions in the higher dimensional center case.