Dimension of invariant measures for affine iterated function systems

Abstract

Let $\{S_i{\}}_{i\in \Lambda }$ be a finite contracting affine iterated function system (IFS) on ${\mathbb{R}}^d$. Let $(\Sigma ,\sigma )$ denote the two-sided full shift over the alphabet $\Lambda$, and $\pi :\Sigma \to {\mathbb{R}}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $\sigma$-invariant measure on $\Sigma$ under $\pi$ 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.

61 pages. Some corrections and clarifications. All the results remain unchanged

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