Dimension of affine iterated function systems on $\mathbb{R}^d$

Abstract

Let $\{f_i(x)=M_ix+a_i\}_{i\in I}$ be an affine iterated function system on $\mathbb{R}^d$ with attractor $\Lambda$, and let $\mu$ be an associated self-affine measure. Assume that the $M_i$ generate a Zariski dense subgroup of $\mathrm{GL}_d(\mathbb{R})$. Under the strong open set condition, we show that $\dim_{\mathrm{H}}\Lambda$ equals the affinity dimension and that $\dim_{\mathrm{H}}\mu$ equals the Lyapunov dimension. The above results are consequences of the following projection theorem, which requires no separation condition. For every $1\leq n\leq d$ and for Furstenberg measure a.e. $(d-n)$-dimensional subspace $V$, \[ \dim_{\mathrm{H}} (P^V)_*\mu=\min\{n,\,\dim_{\mathrm{H}}\mu\}. \] This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures, exploited through Wu's one-dimensional entropy increase theorem. A key ingredient in reducing the higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.