Measure Rigidity and Equidistribution for Fractal Carpets

Abstract Let $\theta $ θ be a Bernoulli measure which is stationary for a random walk generated by finitely many contracting rational affine dilations of $\mathbb{R}^{d}$ R d , and let $\mathcal{K} = \mathrm{supp}(\theta )$ K = supp ( θ ) be the corresponding attractor. An example in dimension $d=1$ d = 1 is the Hausdorff measure on Cantor’s middle thirds set, and examples in higher dimensions include missing digits sets, Sierpiński carpets and Menger sponges. Let $\nu $ ν denote the image of $\theta $ θ under the map $\mathcal{K} \rightarrow \mathrm{SL}_{d+1}(\mathbb{R})/\mathrm{SL}_{d+1}( \mathbb{Z})$ K → SL d + 1 ( R ) / SL d + 1 ( Z ) which sends $\mathbf{x}$ x to the lattice $\Lambda _{\mathbf{x}} = \mathrm{span}_{\mathbb{Z}} (\mathbf{e}_{1}, \ldots , \mathbf{e}_{d}, \mathbf{e}_{d+1}+ (\mathbf{x},0))$ Λ x = span Z ( e 1 , … , e d , e d + 1 + ( x , 0 ) ) . We prove equidistribution of the pushforward measures $a_{n*}\nu $ a n ∗ ν along any diverging sequence of diagonal matrices $(a_{n})\subset \mathrm{SL}_{d+1}(\mathbb{R})$ ( a n ) ⊂ SL d + 1 ( R ) that expand the first $d$ d coordinates under a natural non-escape of mass condition. The latter condition is known to hold whenever $\theta $ θ is absolutely friendly. We also show that weighted badly approximable vectors and Dirichlet-improvable vectors (for arbitrary norm) form a subset of $\mathcal{K}$ K of $\theta $ θ -measure zero. The key ingredient is a measure classification theorem for the stationary measures of an associated random walk on an $S$ S -arithmetic space, introduced by two of the authors in [KL23]. A new feature of this setting is that this random walk admits stationary measures which are not invariant.