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

Let $\{f_i(x)=M_{i}x+a_i{\}}_{i\in I}$ be an affine iterated function system on ${\mathbb{R}}^d$ with attractor $\mathrm{\Lambda }$, and let $\mu$ be an associated self-affine measure. Assume that the $M_i$ generate a Zariski dense subgroup of ${\mathrm{G}\mathrm{L}}_d(\mathbb{R})$. Under the strong open set condition, we show that ${\dim }_{\mathrm{H}}\mathrm{\Lambda }$ 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 $1\le n\le d$ and for Furstenberg measure a.e. $(d-n)$-dimensional subspace $V$, $${\dim }_{\mathrm{H}}(P^V{)}_{\ast }\mu =\min \{n,{\dim }_{\mathrm{H}}\mu \}\mathrm{.}$$ 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.