On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS

TL;DR

This paper investigates the Hausdorff dimension of invariant measures in measurable iterated function systems with place-dependent probabilities, establishing bounds and conditions for ergodicity in a metric space.

Contribution

It provides an upper bound for the Hausdorff dimension of ergodic invariant measures and characterizes ergodicity via the associated skew product.

Findings

Upper bound for Hausdorff dimension of invariant measures.

Ergodicity characterized by the skew product.

Conditions for ergodicity in measurable IFS.

Abstract

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.