Thermodynamic formalism of $GL_2(\mathbb{R})$-cocycles with canonical holonomies

Abstract

We study singular value potentials of H\"older continuous $GL_2(\mathbb{R})$-cocycles over hyperbolic systems whose canonical holonomies converge and are H\"older continuous. Such cocycles include locally constant $GL_2(\mathbb{R})$-cocycles as well as fiber-bunched $GL_2(\mathbb{R})$-cocycles. We show that singular value potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose singular value potentials have more than one equilibrium states.