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Invariance of entropy for maps isotopic to Anosov

Pablo D. Carrasco, Cristina Lizana, Enrique Pujals, Carlos H. Vásquez

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.

Invariance of entropy for maps isotopic to Anosov

Abstract

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

Paper Structure

This paper contains 10 sections, 12 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy
  4. Partial hyperbolicity
  5. Semi-conjugation with the linear part
  6. Structure of the pre-image classes
  7. Bounded Geometry
  8. Robustly transitive example
  9. Acknowledgments
  10. Appendix: proof of Theorem Æ

Key Result

Theorem 1.1

Let $S$ be a compact surface and $f: S\rightarrow S$ be a diffeomorphism in the isotopy class of a pseudo-Anosov $A: S \rightarrow S$. Then $\mathrm{h_{top}}(f) \geq \mathrm{h_{top}}(A)$.

Figures (4)

  • Figure 1: Diagram for the proof.
  • Figure 2: From the left: $h_0$; $h_t,\, 0<t<1/2$; $h_{1/2}$.
  • Figure 3: A zoom inside the disc $\mathbb{D}_{1/2}$.
  • Figure 4: $H$ restricted to $x_1=0$.

Theorems & Definitions (36)

  • Theorem 1.1: Fathi-Shub, Fathi2012
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • ...and 26 more