Thermodynamic formalism of GL₂(ℝ)-cocycles with canonical holonomies

2019/09/25 by Clark Butler, Kiho Park, Butler, Clark +2
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Microtubule and mitosis dynamics #Protein Structure and Dynamics #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.1909.11548

To appear in DCDS

openalex publication_date 2019/09/25 · arxiv created 2020/09/12 · arxiv updated 2020/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study singular value potentials of Hölder continuous GL₂(ℝ)-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant GL₂(ℝ)-cocycles as well as fiber-bunched GL₂(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.

Citations

• Lyapunov exponents with multiplicity 1 for deterministic products of matrices
• Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

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