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Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^{3}$

James Marshall Reber, Sebastián Pavez-Molina

Abstract

Given a $C^0$ conjugacy between two Anosov diffeomorphisms, the matching periodic data problem asks whether this conjugacy is smooth provided spectral data of the diffeomorphisms match at periodic points. We show that if the two $C^0$ conjugate diffeomorphisms on $\mathbb{T}^3$ are sufficiently close to a hyperbolic linear automorphism with a pair of complex conjugate eigenvalues, then the conjugacy must be smooth. In particular, we have that in a neighborhood of a hyperbolic toral automorphism, matching periodic data implies that the conjugacy is $C^{1+\text{Hölder}}$

Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^{3}$

Abstract

Given a conjugacy between two Anosov diffeomorphisms, the matching periodic data problem asks whether this conjugacy is smooth provided spectral data of the diffeomorphisms match at periodic points. We show that if the two conjugate diffeomorphisms on are sufficiently close to a hyperbolic linear automorphism with a pair of complex conjugate eigenvalues, then the conjugacy must be smooth. In particular, we have that in a neighborhood of a hyperbolic toral automorphism, matching periodic data implies that the conjugacy is

Paper Structure

This paper contains 12 sections, 21 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anosov diffeomorphisms
  4. Regularity of foliations
  5. Normal forms
  6. Holonomies and linear cocycles
  7. Periodic cycle holonomies and regularity
  8. Proof of Theorem \ref{['thm:main']}
  9. Parry and Homoclinic Holonomy Groups
  10. Parry monoid and representation
  11. Matching trace alternative
  12. Matching theorem

Key Result

Theorem A

Let $L: \mathbb{T}^{3} \to \mathbb{T}^{3}$ be a hyperbolic automorphism with a pair of non-real complex conjugate eigenvalues and let $r \geq 2$. There is a $C^{1}$-neighborhood $\mathcal{U}$ of $L$ such that if $f,g \in \mathcal{U}$ are any pair of $C^{r}$-Anosov diffeomorphisms which have non-cons

Figures (1)

  • Figure 1: An illustration of the quadrilateral

Theorems & Definitions (35)

  • Definition
  • Theorem A
  • Theorem B
  • Definition
  • Proposition 1
  • Proposition 2
  • proof : Sketch of Proof
  • Proposition 3: sadovskaya2015cohomology
  • Proposition 4
  • Proposition 5
  • ...and 25 more