2017/11/07 by Nikolaos Karaliolios, Karaliolios, Nikolaos
Mathematics · Physics and Astronomy · #37A20 (Primary) 37C05 #46F05 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1711.02732
openalex publication_date 2017/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a general argument for the failure of Anosov-Katok-like constructions (as in \citeAFKo2015 and \citeNKInvDist) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A C∞ smooth diffeomorphism f of a compact manifold M is Cohomologically Rigid iff the equation, known as Linear Cohomological one, ψ∘ f - ψ= φ admits a C∞ smooth solution ψ for every φ in a codimension 1 closed subspace of C∞ (M, ℂ ). As an application, we show that no Cohomologically Rigid diffeomorphisms exist in the Almost Reducibility regime for quasi-periodic cocycles in homogeneous spaces of compact type, even though the Linear Cohomological equation over a generic such system admits a solution for a dense subset of functions φ. We thus confirm a conjecture by M. Herman and A. Katok in that context and provide some insight in the mechanism obstructing the construction of counterexamples.