On the fine structure of stationary measures in systems which contract-on-average

TL;DR

This paper investigates the fine structure of stationary measures for iterated function systems that contract on average, providing bounds on Hausdorff dimension and conditions for measure singularity, with improvements using Shannon's Theorem.

Contribution

It offers new bounds on the Hausdorff dimension of invariant measures and refines conditions for measure singularity in systems contracting on average.

Findings

Upper bound for Hausdorff dimension of invariant measure

Measure is singular if entropy modulus is less than dimension times Lyapunov exponent

Improved estimates using Shannon's Theorem for affine mappings

Abstract

Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a corollary show that the measure will be singular if the modulus of the entropy $\sum_i p_i \log p_i$ is less than $d$ times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of $\mathbb{R}$.