Measure rigidity and equidistribution for fractal carpets
Osama Khalil, Manuel Luethi, Barak Weiss
TL;DR
The paper addresses the problem of understanding how fractal, self-similar Bernoulli measures on irreducible carpet attractors distribute under diverging diagonal flows on the homogeneous space $\mathrm{SL}_{d+1}(\mathbb{R})/\mathrm{SL}_{d+1}(\mathbb{Z})$, with applications to Diophantine approximation. It develops a random-walk framework on an $S$-arithmetic space to classify stationary measures, proving a dichotomy: either the measure is invariant or its stabilizers are trivial, and then deduces equidistribution via an exponential-drift/Ratner-type argument. The main contributions are (i) a measure-classification theorem for stationary measures in this non-stiff, non-invariant setting, (ii) uniform non-divergence results for friendly fractal measures, and (iii) Diophantine consequences showing measure-zero sets for weighted badly approximable and Dirichlet-improvable vectors on the fractals. The work advances the understanding of fractal dynamics on homogeneous spaces and provides tools for transferring equidistribution results to Diophantine properties of typical fractal points, with potential extensions to higher-dimensional fractals and more general norms.
Abstract
Let $θ$ 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 $\mathcal{K} = \mathrm{supp}(θ)$ 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, Sierpiński carpets and Menger sponges. Let $ν$ denote the image of $θ$ under the map $\mathcal{K} \to \mathrm{SL}_{d+1}(\mathbb{R})/\mathrm{SL}_{d+1}(\mathbb{Z})$ which sends $x$ to the lattice $Λ_x = \mathrm{span}_{\mathbb{Z}}(e_1,\ldots,e_d,e_{d+1} + (x,0))$. We prove equidistribution of the pushforward measures $a_{n*}ν$ 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 $θ$ is absolutely friendly. We also show that weighted badly approximable vectors and Dirichlet-improvable vectors (for arbitrary norm) form a subset of $\mathcal{K}$ of $θ$-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.