Under compact support, gap, pinching, and weak-* proximity to volume-preservation, random walks by C² diffeomorphisms on a closed manifold M admit a unique atom-free stationary measure Υ_μ of full Frostman dimension to which μ^{*n} * δ_x converges for most x.
Ergodicity of (co)expanding on average random dynamical systems
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abstract
We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_1,R_2\in \mathrm{SO}(d+1)$, $d\ge 2$, generate a dense subgroup, then the random dynamics of $R_1$ and $R_2$ on $S^d$ is stably ergodic. Previously this was only known to hold in even dimensions. As a consequence, we deduce spectral gap and statistical limit theorems for such systems. In particular, our results apply in the presence of zero Lyapunov exponents.
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2026 1verdicts
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Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime
Under compact support, gap, pinching, and weak-* proximity to volume-preservation, random walks by C² diffeomorphisms on a closed manifold M admit a unique atom-free stationary measure Υ_μ of full Frostman dimension to which μ^{*n} * δ_x converges for most x.