Hyperbolization of cocycles by isometries of the euclidean space

TL;DR

This paper investigates hyperbolized cohomological equations with isometric cocycles in Euclidean space, analyzing their solutions' convergence to genuine solutions and their role as effective approximations.

Contribution

It introduces a hyperbolization approach for cohomological equations with isometric cocycles, providing explicit solutions and convergence analysis.

Findings

Hyperbolized equations have unique continuous solutions.

Solutions serve as good approximations to original equations.

Solutions are global attractors of skew-product dynamics.

Abstract

We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate in to know whether or not these solutions converge to a genuine solution to the original equation, and in what sense we can use them as good approximative solutions. The main advantage of considering solutions to hyperbolized cohomological equations is that they can be easily described, since they are global attractors of a naturally defined skew-product dynamics. We also include some technical results about twisted Birkhoff sums and exponential averaging.