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Long-time stochasticity for periodic iterations of random surface maps
Abstract
Let M be a closed Riemannian surface, and let f1,…,fs be area-preserving C2 diffeomorphisms with fixed positive probabilities such that the resulting random dynamical system is uniformly expanding on average. For each n, choose ω1,…,ωn independently according to these probabilities and set F=fωn∘⋯∘ fω1. We prove that there exists α>0 such that, with probability at least 1−e−αn, the correlations of any two Hölder observables ϕ,ψ under Fm are bounded by e−αn|ϕ|Cβ|ψ|Cβ, for all m≤ eαn. These estimates imply a law of large numbers and a central limit theorem for the Birkhoff sums along F of any zero-mean Hölder observable ϕ, for exponentially long times in n. The limiting Gaussian has variance ∫Mϕ2 dvol.
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BibTeX
@misc{dolgopyat2026longtime,
title = {{Long-time stochasticity for periodic iterations of random surface maps}},
author = {Dmitry Dolgopyat and Bassam Fayad and Alexey Okunev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42083},
eprint = {2610.42083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42083},
}
RIS
TY - UNPB
TI - Long-time stochasticity for periodic iterations of random surface maps
AU - Dmitry Dolgopyat
AU - Bassam Fayad
AU - Alexey Okunev
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42083
UR - https://arxiv.org/abs/2610.42083
AB - Let M be a closed Riemannian surface, and let f_1,…,f_s be area-preserving C^2 diffeomorphisms with fixed positive probabilities such that the resulting random dynamical system is uniformly expanding on average. For each n, choose ω_1,…,ω_n independently according to these probabilities and set F=f_ω_n∘⋯∘ f_ω_1. We prove that there exists α>0 such that, with probability at least 1−e^−αn, the correlations of any two Hölder observables ϕ,ψ under F^m are bounded by e^−αn|ϕ|_C^β|ψ|_C^β, for all m≤ e^αn. These estimates imply a law of large numbers and a central limit theorem for the Birkhoff sums along F of any zero-mean Hölder observable ϕ, for exponentially long times in n. The limiting Gaussian has variance ∫_Mϕ^2 dvol.
ER -