References & Citations
Mathematics > Dynamical Systems
Title: Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner
(Submitted on 5 Jul 2021 (this version), latest version 9 Mar 2022 (v2))
Abstract: We develop a new method to study ergodic integrals for horocycle flows which does not rely on the study of the cohomological equation. Our approach is inspired by Ratner's work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987). We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni (Duke Math. J., 2003), and we show that the coefficients in the asymptotic expansion are H\"{o}lder continuous with respect to the base point. Furthermore, we provide short and self-contained proofs of the spatial limit theorems of Bufetov and Forni (Ann. Sci. \'Ec. Norm. Sup\'er., 2014) and, in an appendix by Emilio Corso, of a temporal limit theorem by Dolgopyat and Sarig (J. Stat. Phys., 2017).
Submission history
From: Davide Ravotti [view email][v1] Mon, 5 Jul 2021 15:30:14 GMT (23kb)
[v2] Wed, 9 Mar 2022 14:20:10 GMT (25kb)
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