Self-similar measures associated to a homogeneous system of three maps

TL;DR

This paper investigates the dimension of self-similar measures generated by a system of three contracting maps, extending recent Bernoulli convolution theories and developing new techniques to address unique phenomena.

Contribution

It introduces novel methods, including an extension of Hochman's entropy increase technique, to analyze self-similar measures for three-map systems, revealing new phenomena.

Findings

Identification of new phenomena in three-map systems

Extension of Hochman's entropy method to function fields

Results on the dimension of self-similar measures

Abstract

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and B\'ar\'any, K\"aenm\"aki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup.