Measure rigidity and equidistribution for fractal carpets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910847420334080 |
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| author | Khalil, Osama Luethi, Manuel Weiss, Barak |
| author_facet | Khalil, Osama Luethi, Manuel Weiss, Barak |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure rigidity and equidistribution for fractal carpets Khalil, Osama Luethi, Manuel Weiss, Barak Dynamical Systems Number Theory 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. |
| title | Measure rigidity and equidistribution for fractal carpets |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2502.19552 |