Measure rigidity and equidistribution for fractal carpets

Abstract

Let $\theta$ be a Bernoulli measure which is stationary for a random walk generated by finitely many contracting rational affine dilations of $\mathbb{R}^d$, and let $\mathcal{K} = \mathrm{supp}(\theta)$ be the corresponding attractor. An example in dimension $d=1$ is the Hausdorff measure on Cantor's middle thirds set, and examples in higher dimensions include missing digits sets, Sierpi\'nski carpets and Menger sponges. Let $\nu$ denote the image of $\theta$ under the map $\mathcal{K} \to \mathrm{SL}_{d+1}(\mathbb{R})/\mathrm{SL}_{d+1}(\mathbb{Z})$ which sends $x$ to the lattice $\Lambda_x = \mathrm{span}_{\mathbb{Z}}(e_1,\ldots,e_d,e_{d+1} + (x,0))$. We prove equidistribution of the pushforward measures $a_{n*}\nu$ along any diverging sequence of diagonal matrices $(a_n)\subset\mathrm{SL}_{d+1}(\mathbb{R})$ that expand the first $d$ coordinates under a natural non-escape of mass condition. The latter condition is known to hold whenever $\theta$ is absolutely friendly. We also show that weighted badly approximable vectors and Dirichlet-improvable vectors (for arbitrary norm) form a subset of $\mathcal{K}$ of $\theta$-measure zero. The key ingredient is a measure classification theorem for the stationary measures of an associated random walk on an $S$-arithmetic space, introduced by the two first-named authors in earlier work. A new feature of this setting is that this random walk admits stationary measures which are not invariant.