Density of non-zero exponent of contraction for pinching cocycles on Hom(S^1)

TL;DR

This paper demonstrates that in the setting of circle homeomorphisms over hyperbolic bases, cocycles with non-zero contraction exponents are densely prevalent, extending linear cocycle results to a non-linear, non-differentiable context.

Contribution

It generalizes the concept of contraction exponents and their density from linear and differentiable cocycles to non-linear, non-differentiable cocycles on the circle.

Findings

Non-zero contraction exponents are dense among the considered cocycles.

Extension of linear cocycle results to non-linear, non-differentiable cases.

Application of Malicet's Invariance Principle to this setting.

Abstract

We consider pinching cocycles taking values in the space of homeomorphisms of the circle over an hyperbolic base. Using the Invariance Principle of Malicet, we prove that the cocycles having non-zero exponents of contraction are dense. In this article we generalize some common notions an results known of linear cocycles and cocycles of diffeomorphisms, to the non-linear non-differentiable case.