Thermodynamic formalism of $  \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies

Clark Butler; Kiho Park

doi:10.3934/dcds.2020356

Article Contents

2021, Volume 41, Issue 5: 2141-2166. Doi: 10.3934/dcds.2020356

This issue Previous Article Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems Next Article Radially symmetric stationary wave for two-dimensional Burgers equation

Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies

Clark Butler^1, and
Kiho Park^2,

1.
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
2.
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

Received: January 31, 2020

Revised: July 31, 2020

Early access: October 2020

Published: May 2021

The first author is supported by Oswald Veblen fund

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Abstract

We study the norm potentials of Hölder continuous $ \text{GL}_2(\mathbb{R}) $-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant $ \text{GL}_2(\mathbb{R}) $-cocycles as well as fiber-bunched $ \text{GL}_2(\mathbb{R}) $-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states.

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Mathematics Subject Classification: 37D35, 37C40, 37H15.

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