Thermodynamic formalism for invariant measures in iterated function systems with overlaps

TL;DR

This paper develops a thermodynamic formalism for invariant measures in conformal iterated function systems with overlaps, providing a dimension formula and conditions for dimension drop based on overlap numbers.

Contribution

It introduces a geometric dimension formula for self-conformal measures with overlaps and characterizes when dimension drops occur.

Findings

Exact dimensionality of image measures established

Dimension formula involving overlap numbers derived

Necessary and sufficient condition for dimension drop identified

Abstract

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for iterated function systems with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $\nu = \pi_*\mu$ is a self-conformal measure, then $HD(\nu) < \frac{h(\mu)}{|\chi(\mu)|}$ if and only if the overlap number $o(\mathcal S, \mu) > 1$. Examples are also discussed.