Dimension of invariant measures for affine iterated function systems

Abstract

Let ${\{S_i\}}_{i\in \mathrm{\Lambda }}$ be a finite contracting affine iterated function system (IFS) on ${\mathbb{R}}^d$. Let $(\mathrm{\Sigma },\mathit{\sigma })$ denote the two-sided full shift over the alphabet Λ, and let $\mathrm{\pi }:\mathrm{\Sigma }\to {\mathbb{R}}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier–Young-type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.