On the cohomology class of fiber-bunched cocycles on semi simple Lie groups
type: Item
instance of: scholarly article
title: On the cohomology class of fiber-bunched cocycles on semi simple Lie groups
zbMATH Open document ID: 1462.37061
author: Paulo Varandas
published in: Revista Matemática Iberoamericana
publication date: 2021-03-07T00:00:00Z
review text: Summary: We study the cohomological equation associated to linear cocycles on semi simple Lie groups \(\mathcal{G}\) over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with \(C^\nu(M,\mathcal{G})\). In particular, we prove that there exists an open and dense subset of the set \(C_b^\nu(M,\mathcal{G})\) of fiber-bunched cocycles with trivial centralizer. As a consequence we deduce that the solutions of the cohomological equation of fiber-bunched cocycles form a finite abelian subgroup of \(C_b^\nu(M,\mathcal{G})\) for an open and dense subset of fiber-bunched cocycles in \(C_b^\nu(M,\mathcal{G})\). Some results on the centralizer of skew-products are also given.
Mathematics Subject Classification ID: 37H15
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 7318478
zbMATH Keywords: cohomological equation
zbMATH Keywords: fiber-bunched cocycles
zbMATH Keywords: centralizers
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/rmi/1160
OpenAlex ID: W2997383508
Wikidata QID: Q115211920
cites work: Rigidity of fiber bunched cocycles
cites work: Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases
cites work: Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices. (Genericity of non-zero Lyapunov exponents for deterministic products of matrices)
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Centralizers of partially hyperbolic diffeomorphisms
cites work: ROOTS AND CENTRALIZERS OF ANOSOV DIFFEOMORPHISMS ON TORI
cites work: Lyapunov exponents of linear cocycles. Continuity via large deviations
cites work: Q5614003
cites work: Livšic theorem for matrix cocycles
cites work: Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms
cites work: Q5631808
cites work: Q5634639
cites work: COHOMOLOGY OF DYNAMICAL SYSTEMS
cites work: Livšic's theorem for semisimple Lie groups
cites work: Regularity of the transfer map for cohomologous cocycles
cites work: The Livsic Cocycle Equation for Compact Lie Group Extensions of Hyperbolic Systems
cites work: Livsic theorems for connected Lie groups
cites work: On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms
cites work: The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial
cites work: Cohomology of fiber bunched cocycles over hyperbolic systems
cites work: Remarks on Livšic' theory for nonabelian cocycles
cites work: Lectures on Lyapunov Exponents
cites work: Solutions to the twisted cocycle equation over hyperbolic systems
DOI: 10.4171/RMI/1160
description: scientific article; zbMATH DE number 7318478