Dimension theory of iterated function systems

Abstract

Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$ on the space of invariant measures on $\Sigma$ associated with the coding map $\pi: \Sigma\to K$, and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on $K$. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under $\pi$ is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures.