Overlap functions for measures in conformal iterated function systems

Abstract

We study conformal iterated function systems (IFS) $\mathcal S = \{ϕ_i\}_{i \in I}$ with arbitrary overlaps, and measures $μ$ on limit sets $Λ$, which are projections of equilibrium measures $\hat μ$ with respect to a certain lift map $Φ$ on $Σ_I^+ \times Λ$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat μ$ 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 $o(\mathcal S, \hat μ)$ is related to a conditional entropy of $\hat μ$ with respect to the lift $Φ$. Various types of projections to $Λ$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(μ)$ of $μ$ on $Λ$, by using pressure functions and $o(\mathcal S, \hat μ)$. In particular, this applies to projections of Bernoulli measures on $Σ_I^+$. Next, we apply the results to Bernoulli convolutions $ν_λ$ for $λ\in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_λ$. We prove that for all $λ\in (\frac 12, 1)$, there exists a relation between $HD(ν_λ)$ and the overlap number $o(\mathcal S_λ)$. The number $o(\mathcal S_λ)$ is approximated with integrals on $Σ_2^+$ with respect to the uniform Bernoulli measure $ν_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_λ)$ for certain values of $λ$.