Livšic Theorems for Non-Commutative Groups including Diffeomorphism Groups and Results on the Existence of Conformal Structures for Anosov Systems

2007/11/20 by de la Llave, Rafael, Windsor, Alistair · 1 citation
#37C40 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.0711.3229

Abstract

The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov diffeomorphism f on M and a Holder function η: M ↦ G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M ↦ G satisfying η(x) = ϕ(f(x)) ϕ(x)^-1 that a condition -- obviously necessary -- on the cocycle generated by ηrestricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.

Cited by

• Transitivity and an abelian Livsic theorem for covers

Related

• Livsic theorem for diffeomorphism cocycles
• Livsic theorem for low-dimensional diffeomorphism cocycles
• On Livsic's Theorem, Superrigidity, and Anosov Actions of Semisimple Lie Groups
• Linear cocycles over hyperbolic systems and criteria of conformality
• Transitivity and an abelian Livsic theorem for covers