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Vortices over Riemann surfaces and dominated splittings

Thomas Mettler, Gabriel P. Paternain

Abstract

We associate a flow $φ$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $φ$ always admits a dominated splitting and identify special cases in which $φ$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

Vortices over Riemann surfaces and dominated splittings

Abstract

We associate a flow to a solution of the vortex equations on a closed oriented Riemannian 2-manifold of negative Euler characteristic and investigate its properties. We show that always admits a dominated splitting and identify special cases in which is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of .

Paper Structure

This paper contains 25 sections, 14 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Vortices
  4. Vortex thermostats
  5. Results
  6. Preliminaries
  7. The unit tangent bundle
  8. Roots of the unit tangent bundle
  9. Thermostats
  10. Dominated splittings and hyperbolicity
  11. Definitions
  12. The derivative cocycle of a thermostat
  13. Infinitesimal generators and conjugate cocycles
  14. Conditions ensuring domination and hyperbolicity
  15. Bi-contact structures
  16. ...and 10 more sections

Key Result

Theorem A

Every vortex thermostat admits a dominated splitting. More-over, if all closed orbits of $\phi$ are hyperbolic saddles, then $\phi$ is Anosov.

Figures (1)

  • Figure 1: Dominated splitting property

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Remark 2.1: Notation
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Remark 3.4
  • ...and 27 more