A Livsic type theorem for germs of analytic diffeomorphisms

TL;DR

This paper explores the extension of Livsic's theorem to cocycles of germs of analytic diffeomorphisms over hyperbolic systems, providing positive results under certain conditions.

Contribution

It establishes a Livsic type theorem for germs of analytic diffeomorphisms, advancing understanding of cocycle cohomology in hyperbolic dynamics.

Findings

Proves Livsic's theorem for germs of analytic diffeomorphisms.

Addresses orbit periodic obstructions in hyperbolic systems.

Provides conditions under which the theorem holds.

Abstract

We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.