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Product Anosov diffeomorphisms and the two-sided limit shadowing property

Bernardo Carvalho

Abstract

We characterize product Anosov diffeomorphisms in terms of the two-sided limit shadowing property. It is proved that an Anosov diffeomorphism is a product Anosov diffeomorphism if and only if any lift to the universal covering has the unique two-sided limit shadowing property. Then we introduce two maps in a suitable Banach space such that fixed points of these maps are related with shadowing orbits on the universal covering.

Product Anosov diffeomorphisms and the two-sided limit shadowing property

Abstract

We characterize product Anosov diffeomorphisms in terms of the two-sided limit shadowing property. It is proved that an Anosov diffeomorphism is a product Anosov diffeomorphism if and only if any lift to the universal covering has the unique two-sided limit shadowing property. Then we introduce two maps in a suitable Banach space such that fixed points of these maps are related with shadowing orbits on the universal covering.

Paper Structure

This paper contains 5 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Lifting shadowing properties
  3. The general scenario
  4. The Anosov scenario
  5. Proof of Theorem \ref{['bolado']}

Key Result

Theorem 1.1

An Anosov diffeomorphism on a compact and connected manifold is transitive if and only if it has the two-sided limit shadowing property.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Remark
  • Remark
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 10 more