Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups

2008/09/08 by Rafael de la Llave, de la Llave, Rafael, Alistair Windsor +1
Computer Science · Mathematics · #37D20 #58D05 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #advanced mathematical theories #math.DS #msc:37D20 #msc:58D05

paper · pdf · doi:10.48550/arxiv.0809.1235

arxiv created 2008/09/08 · openalex publication_date 2008/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let f be a transitive Anosov diffeomorphism of a compact manifold M. Suppose that η∈ C^\reg(M,\Diff^r(N)) for a compact manifold N, k,r ∈ \N, r ≥ 1, and 0 < α≤ \Lip. We show that if there exists a φ∈ C^\reg(M,\Diff¹(N)) solving φ_f(x) = η_x ∘ φ_x then in fact φ∈ C^\reg(M,\Diff^r(N)).

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