On the dimension theory of random walks and group actions by circle diffeomorphisms

TL;DR

This paper investigates the dimensional properties of measures and invariant sets related to random walks and group actions by circle diffeomorphisms, extending classical results and introducing new theoretical tools.

Contribution

It introduces a structure theorem for smooth random walks on the circle and generalizes the critical exponent concept of Fuchsian groups to dynamical systems.

Findings

Minimal exceptional sets have Hausdorff dimension less than one.

The Hausdorff dimension of minimal sets containing fixed points exceeds a specific bound.

Generalization of classical Fuchsian group results to non-linear circle actions.

Abstract

We establish new results on the dimensional properties of measures and invariant sets associated to random walks and group actions by circle diffeomorphisms. This leads to several dynamical applications. Among the applications, we show, strengthening of a recent result of Deroin-Kleptsyn-Navas [24], that the minimal set of a finitely generated group of real-analytic circle diffeomorphisms, if exceptional, must have Hausdorff dimension less than one. Moreover, if the minimal set contains a fixed point of multiplicity k + 1 of an diffeomorphism of the group, then its Hausdorff dimension must be greater than k/(k + 1). These results generalize classical results about Fuchsian group actions on the circle to non-linear settings. This work is built on three novel components, each of which holds its own interest: a structure theorem for smooth random walks on the circle, several dimensional properties of smooth random walks on the circle and a dynamical generalization of the critical exponent of Fuchsian groups.