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

Abstract

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\mathbf{R}$ 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 Bárány, Käenmäki. 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.

arXiv comment85 pages, minor revision based on referee's report, final accepted version to appear in Duke Math. J