Liv\v{s}ic Theorem for Banach Rings

TL;DR

This paper extends the Livšic Theorem to cocycles with values in Banach rings, establishing conditions under which such cocycles are coboundaries with H"older continuous transfer functions in a dynamical systems setting.

Contribution

It proves the Livšic Theorem for H"older continuous cocycles valued in Banach rings, broadening the theorem's applicability to more general algebraic structures.

Findings

Established the equivalence between periodic orbit conditions and coboundary property for Banach ring cocycles.

Extended Livšic Theorem to include Banach ring-valued cocycles with H"older continuity.

Provided conditions under which cocycles are trivialized by H"older continuous transfer functions.

Abstract

We prove the Liv\v{s}ic Theorem for H\"{o}lder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${\sigma:X\to X}$ that satisfies the Anosov Closing Lemma, and a H\"{o}lder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a H\"{o}lder continuous transition function if and only if ${a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)=e}$ for each periodic point $p=\sigma^n p$.