# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020356

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

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

Received  February 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by Oswald Veblen fund

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.

Citation: Clark Butler, Kiho Park. Thermodynamic formalism of $\text{GL}_2(\mathbb{R})$-cocycles with canonical holonomies. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020356
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##### References:
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