Livšic Theorem for Banach Rings

2014/08/24 by Genady Ya. Grabarnik, Grabarnik, Genady Ya., Misha Guysinsky +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS

paper · pdf · doi:10.48550/arxiv.1408.5639

arxiv created 2014/08/24 · openalex publication_date 2014/08/24 · arxiv updated 2014/08/26 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

We prove the Livšic Theorem for Hölder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism σ:X→ X that satisfies the Anosov Closing Lemma, and a Hölder continuous map a:X→ B^× from a compact metric space X to the set of invertible elements of some Banach ring B. We show that it is a coboundary with a Hölder continuous transition function if and only if a(σ^n-1p)… a(σp)a(p)=e for each periodic point p=σⁿ p.

Citations

• Products of Random Matrices

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