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Vortices over Riemann surfaces and dominated splittings

Part of: Dynamical systems with hyperbolic behavior Riemann surfaces

Published online by Cambridge University Press:  02 February 2021

THOMAS METTLER Open the ORCID record for THOMAS METTLER [Opens in a new window]  and
GABRIEL P. PATERNAIN
THOMAS METTLER*
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt am Main, Germany (e-mail: mettler@math.ch)
GABRIEL P. PATERNAIN
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (e-mail: g.p.paternain@dpmms.cam.ac.uk)
*
e-mail: mettler@math.uni-frankfurt.de
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Abstract

We associate a flow $\phi $ with 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 $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ 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 $(M,g)$.

Keywords

thermostatsvortex equationsAnosov flowsdominated splittingfractional differentials

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 30F30: Differentials on Riemann surfaces

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1781 - 1806
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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