Vortices over Riemann Surfaces and Dominated Splittings

Thomas Mettler

Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt am Main, Germany

Gabriel P. Paternain

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, England

Date: 27th November 2020

Published Version: Ergodic Theory Dynam. Systems 42 (2022), 1781—1806

1 Introduction
1.1 Background
1.2 Vortices
1.3 Vortex thermostats
1.4 Results
2 Preliminaries
2.1 The unit tangent bundle
2.2 Roots of the unit tangent bundle
2.3 Thermostats
3 Dominated splittings and hyperbolicity
3.1 Definitions
3.2 The derivative cocycle of a thermostat
3.3 Infinitesimal generators and conjugate cocycles
3.4 Conditions ensuring domination and hyperbolicity
3.5 Bi-contact structures
4 Thermostats from Vortices
4.1 The vortex equations
4.2 The vortex equations on a root of \(SM\)
4.3 Fractional differentials
4.4 The thermostat
5 Proof of Theorems A and B
5.1 Dominated splitting
5.2 The Anosov property
6 Examples
Appendix A Variants of the vortex equations
References