Invariance of entropy for maps isotopic to Anosov

TL;DR

This paper proves that the topological entropy remains invariant for certain partially hyperbolic diffeomorphisms on tori with simple central bundles and hyperbolic induced action, providing a counterexample when the simplicity condition is absent.

Contribution

It establishes entropy invariance under specific conditions and constructs a robust counterexample without the simplicity assumption.

Findings

Entropy is invariant for maps with simple central bundles and hyperbolic induced action.

Counterexample shows entropy can vary without the simplicity condition.

Provides a deeper understanding of entropy behavior in partially hyperbolic systems.

Abstract

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on $H_1(\mathbb{T}^d)$ is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example.