Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime

Abstract

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 $\mathbb{S}^d$.