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A note on the density of periodic orbits of Anosov geodesic flow in manifolds of finite volume

Nestor Nina Zarate, Sergio Romaña

Abstract

In this paper, we prove that manifolds of finite volume with Anosov geodesic flow have dense periodic orbits. The same result works for conservative Anosov flows in non-compact cases.

A note on the density of periodic orbits of Anosov geodesic flow in manifolds of finite volume

Abstract

In this paper, we prove that manifolds of finite volume with Anosov geodesic flow have dense periodic orbits. The same result works for conservative Anosov flows in non-compact cases.
Paper Structure (7 sections, 10 theorems, 59 equations)

This paper contains 7 sections, 10 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable and Unstable Manifolds
  4. Local Product Structure
  5. Shadowing Lemma
  6. Proof of the Theorem \ref{['teo2.7']}
  7. Proof of Corollary \ref{['C1-Main-T']}

Key Result

Theorem 1.1

If $M$ has finite volume and $\phi^t:SM\to SM$ is an Anosov geodesic flow, then the periodic orbits of $\phi^t$ are dense in $SM$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Theorem 2.2: Local Product Structure
  • Definition 3.1: Shadowing forward and backward
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 9 more