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 R and other more general iterated function systems. 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 as well as Bárány and Käenmäki. To overcome the difficulties stemming from this phenomenon, we develop novel techniques, including an extension of Hochman’s entropy increase method to a function field setup.