Multislicing and effective equidistribution for random walks on some homogeneous spaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Bénard, Timothée He, Weikun |
| author_facet | Bénard, Timothée He, Weikun |
| contents | We consider a random walk on a homogeneous space $G/Λ$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x \in G/Λ$ which is not trapped in a finite $μ$-invariant set, the $n$-step distribution $μ^{*n}*δ_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit.
Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems à la Bourgain and we believe it presents independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multislicing and effective equidistribution for random walks on some homogeneous spaces Bénard, Timothée He, Weikun Dynamical Systems Classical Analysis and ODEs Primary 37A99, Secondary 22E99, 51B99, 60G50 We consider a random walk on a homogeneous space $G/Λ$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x \in G/Λ$ which is not trapped in a finite $μ$-invariant set, the $n$-step distribution $μ^{*n}*δ_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems à la Bourgain and we believe it presents independent interest. |
| title | Multislicing and effective equidistribution for random walks on some homogeneous spaces |
| topic | Dynamical Systems Classical Analysis and ODEs Primary 37A99, Secondary 22E99, 51B99, 60G50 |
| url | https://arxiv.org/abs/2409.03300 |