Dimension of invariant measures for affine iterated function systems

TL;DR

This paper proves that ergodic invariant measures for affine iterated function systems have exact dimensional projections with a Hausdorff dimension given by a Ledrappier-Young type formula, extending previous results and addressing an open question.

Contribution

It establishes the exact dimensionality and dimension formula for projections of ergodic measures in affine IFSs, including average contracting systems, resolving longstanding open problems.

Findings

Projections of ergodic measures are exact dimensional.

Hausdorff dimension follows a Ledrappier-Young type formula.

Results apply to self-affine sets and measures.

Abstract

Let $\{S_i\}_{i\in \Lambda}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(\Sigma,\sigma)$ denote the two-sided full shift over the alphabet $\Lambda$, and $\pi:\Sigma\to {\Bbb R}^d$ 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.