Dimension of contracting on average self-similar measures

TL;DR

This paper extends Hochman's theorem to a broader class of self-similar measures, introducing a new variance summation method that relaxes previous separation conditions and simplifies the proof.

Contribution

It generalizes the dimension results for self-similar measures to contracting on average measures using a novel variance summation technique.

Findings

Dimension of contracting on average measures is characterized similarly to classical self-similar measures.

We demonstrate that exponential separation is not necessary for dimension results.

The variance summation method provides a new approach avoiding inverse entropy theorems.

Abstract

We generalise Hochman's theorem on the dimension of self-similar measures to contracting on average measures and show that a weaker condition than exponential separation on all scales is sufficient. Our proof uses a technique we call the variance summation method, avoiding the use of inverse theorems for entropy.