Thermodynamic formalism for invariant measures in iterated function systems with overlaps

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 $\nu ={\pi }_{\ast }\mu$ 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.

To appear in Communications in Contemporary Mathematics, DOI: 10.1142/S0219199721500413. arXiv admin note: substantial text overlap with arXiv:1908.10050