Typical absolute continuity for classes of dynamically defined measures

TL;DR

This paper investigates the absolute continuity and dimensional properties of measures generated by parameter-dependent iterated function systems, introducing new methods to handle the variable measures and establishing conditions for almost sure absolute continuity.

Contribution

It develops a novel approach for analyzing the absolute continuity of projected measures when the measures depend on parameters, extending previous fixed-measure frameworks.

Findings

Projected measures are absolutely continuous for almost every parameter under certain entropy and Lyapunov ratio conditions.

Provides an almost sure lower bound on Sobolev dimension for parameter-dependent measures.

Derives an almost sure formula for Hausdorff dimension under less restrictive assumptions.

Abstract

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^\lambda_j\}$ on the real line. Given a family of parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $\Pi^\lambda$. The main novelty of our work is that the measures $\mu_\lambda$ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure $(\Pi^\lambda)_*\mu_\lambda$, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if $\mu_\lambda$ are Gibbs measures for a family of H\"older continuous potentials $\phi^\lambda$, with H\"older continuous dependence on $\lambda$ and $\{\Pi^\lambda\}$ satisfy the transversality condition, then the projected measure $(\Pi^\lambda)_*\mu_\lambda$ is absolutely continuous for Lebesgue a.e.\ $\lambda$, such that the ratio of entropy over the Lyapunov exponent is strictly greater than $1$. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).