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Advances on Stable Ergodicity of Toral Automorphisms

Fernando Argentieri, Andrea Ulliana

Abstract

We prove that all ergodic automorphisms of the $N$-dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension $N\leq 5$ or $N=7$. This generalizes a previous result of Rodriguez-Hertz, that required an additional algebraic condition on the carachteristic polynomial of the linear automorphism. The core of the proof is a minimality criterion.

Advances on Stable Ergodicity of Toral Automorphisms

Abstract

We prove that all ergodic automorphisms of the -dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension or . This generalizes a previous result of Rodriguez-Hertz, that required an additional algebraic condition on the carachteristic polynomial of the linear automorphism. The core of the proof is a minimality criterion.
Paper Structure (16 sections, 36 theorems, 30 equations)

This paper contains 16 sections, 36 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Background and Organization
  3. Partial Hyperbolicity
  4. Lifts and Perturbations
  5. Plan of Proof and Structure of the Paper
  6. Geometric Preliminaries
  7. Invariant Manifolds and Holonomies
  8. Saturation
  9. Algebraic preliminaries
  10. Characteristic Polynomial Properties
  11. Diophantine Estimates
  12. Proof of the Minimality Criterion
  13. The structure of U
  14. Conclusion of the proof
  15. Proof of Stable Ergodicity
  16. ...and 1 more sections

Key Result

Theorem 1

Any ergodic linear automorphism $A$ of the torus ${T}^N$ with $\dim(E^c)=2$ is stably ergodic in $\mathcal{C}^{22}_{\text{vol}}$.

Theorems & Definitions (62)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 52 more