Overlap Functions for Measures in Conformal Iterated Function Systems

Abstract

We study conformal iterated function systems (IFS) $\mathcal S = \{\phi_i\}_{i \in I}$ with arbitrary overlaps, and measures $\mu$ on limit sets $\Lambda$, which are projections of equilibrium measures $\hat \mu$ with respect to a certain lift map $\Phi$ on $\Sigma_I^+ \times \Lambda$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat \mu$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\mathcal S, \hat \mu)$ is related to a conditional entropy of $\hat \mu$ with respect to the lift $\Phi$. Various types of projections to $\Lambda$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(\mu)$ of $\mu$ on $\Lambda$, by using pressure functions and $o(\mathcal S, \hat \mu)$. In particular, this applies to projections of Bernoulli measures on $\Sigma_I^+$. Next, we apply the results to Bernoulli convolutions $\nu_\lambda$ for $\lambda \in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_\lambda$. We prove that for all $\lambda \in (\frac 12, 1)$, there exists a relation between $HD(\nu_\lambda)$ and the overlap number $o(\mathcal S_\lambda)$. The number $o(\mathcal S_\lambda)$ is approximated with integrals on $\Sigma_2^+$ with respect to the uniform Bernoulli measure $\nu_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_\lambda)$ for certain values of $\lambda$.