Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916050009849856 |
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| author | Bénard, Timothée Zhang, Zhiyuan |
| author_facet | Bénard, Timothée Zhang, Zhiyuan |
| contents | We consider a random walk on a closed manifold $M$ driven by a probability measure $μ$ on the space of $C^2$ diffeomorphisms. Provided $μ$ has compact support, satisfies certain gap and pinching conditions, and is weak-$*$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure $Υ_μ$. This measure has full Frostman dimension and coincides with volume in the volume-preserving setting. Moreover, for every $x\in M$, the $n$-step distribution $μ^{*n} * δ_x$ converges to $Υ_μ$ unless $x$ is contained in a finite $μ$-invariant set. Our result applies to a variety of situations, including bi-expanding random walks on surfaces, non-linear perturbations of Zariski-dense random walks on the torus $\mathbb{T}^d$, on cocompact lattice quotients of $\mathrm{SO}(2,1)$ and $\mathrm{SO}(3,1)$, and on the sphere $\mathbb{S}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27008 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime Bénard, Timothée Zhang, Zhiyuan Dynamical Systems We consider a random walk on a closed manifold $M$ driven by a probability measure $μ$ on the space of $C^2$ diffeomorphisms. Provided $μ$ has compact support, satisfies certain gap and pinching conditions, and is weak-$*$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure $Υ_μ$. This measure has full Frostman dimension and coincides with volume in the volume-preserving setting. Moreover, for every $x\in M$, the $n$-step distribution $μ^{*n} * δ_x$ converges to $Υ_μ$ unless $x$ is contained in a finite $μ$-invariant set. Our result applies to a variety of situations, including bi-expanding random walks on surfaces, non-linear perturbations of Zariski-dense random walks on the torus $\mathbb{T}^d$, on cocompact lattice quotients of $\mathrm{SO}(2,1)$ and $\mathrm{SO}(3,1)$, and on the sphere $\mathbb{S}^d$. |
| title | Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.27008 |