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Anosov magnetic flows on surfaces

James Marshall Reber, Yumin Shen

Abstract

Using the quotient bundle introduced by Wojtkowski, we give necessary and sufficient conditions for a magnetic flow on a closed, oriented surface to be Anosov.

Anosov magnetic flows on surfaces

Abstract

Using the quotient bundle introduced by Wojtkowski, we give necessary and sufficient conditions for a magnetic flow on a closed, oriented surface to be Anosov.

Paper Structure

This paper contains 7 sections, 26 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The quotient bundle
  3. Proof of Theorem \ref{['thm:conjugate']}
  4. Green bundles
  5. Comparison theory
  6. Stability criteria
  7. Proof of Theorem \ref{['thm:main']} and Corollary \ref{['cor:main2']}

Key Result

Theorem 1

Let $M$ be a closed, oriented surface, and let $(g,b)$ be a magnetic system on $M$. If ${\varphi_{g,b}^t} : SM \rightarrow SM$ is Anosov, then $(g,b)$ is without conjugate points.

Theorems & Definitions (45)

  • Theorem 1
  • Corollary 2
  • Remark 1
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Lemma 2
  • Lemma 3
  • proof
  • ...and 35 more