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Mixing asymptotics for time-changes of horocycle flows

Davide Ravotti

TL;DR

This work delivers sharp, polynomial mixing rates for smooth time-changes of horocycle flows on compact hyperbolic surfaces. By coupling a refined mixing-via-shearing argument with a detailed ergodic-integral expansion that tracks spectral data of the Casimir operator, the authors obtain polynomial decay bounds matching Ratner’s rates for the unperturbed flow and, when a spectral gap below 1/4 is present, exact polynomial asymptotics. The analysis hinges on precise control of ergodic integrals, the distortion of geodesic push-forwards, and a nuanced treatment of principal-series contributions. The results significantly advance understanding of parabolic dynamics under perturbations and establish robust, sharp decay rates for a broad class of smooth time-changes.

Abstract

Mixing-via-shearing is a powerful and versatile method for establishing mixing properties of smooth parabolic flows. In its quantitative form, it provides upper bounds on the decay of correlations for sufficiently smooth observables. Despite its wide applicability, determining the exact rates of mixing for a given smooth parabolic flow remains notoriously difficult. Apart from the classical horocycle flow, no examples are known where polynomial asymptotics, or sharp lower bounds, hold. In this paper, we address this question for smooth time-changes of horocycle flows on compact hyperbolic surfaces. Our approach relies on a refined version of the mixing-via-shearing method which leverages on a precise description of the ergodic integrals for horocycle flows, in particular of the regularity of the coefficients appearing in their asymptotic expansions. Using this method, we prove polynomial upper bounds on the decay of correlations for smooth observables that match the optimal rates originally obtained by Ratner for the standard horocycle flow. Furthermore, in the presence of a spectral gap below $1/4$, we establish exact polynomial asymptotics, mirroring the classical behavior of the horocycle flow.

Mixing asymptotics for time-changes of horocycle flows

TL;DR

This work delivers sharp, polynomial mixing rates for smooth time-changes of horocycle flows on compact hyperbolic surfaces. By coupling a refined mixing-via-shearing argument with a detailed ergodic-integral expansion that tracks spectral data of the Casimir operator, the authors obtain polynomial decay bounds matching Ratner’s rates for the unperturbed flow and, when a spectral gap below 1/4 is present, exact polynomial asymptotics. The analysis hinges on precise control of ergodic integrals, the distortion of geodesic push-forwards, and a nuanced treatment of principal-series contributions. The results significantly advance understanding of parabolic dynamics under perturbations and establish robust, sharp decay rates for a broad class of smooth time-changes.

Abstract

Mixing-via-shearing is a powerful and versatile method for establishing mixing properties of smooth parabolic flows. In its quantitative form, it provides upper bounds on the decay of correlations for sufficiently smooth observables. Despite its wide applicability, determining the exact rates of mixing for a given smooth parabolic flow remains notoriously difficult. Apart from the classical horocycle flow, no examples are known where polynomial asymptotics, or sharp lower bounds, hold. In this paper, we address this question for smooth time-changes of horocycle flows on compact hyperbolic surfaces. Our approach relies on a refined version of the mixing-via-shearing method which leverages on a precise description of the ergodic integrals for horocycle flows, in particular of the regularity of the coefficients appearing in their asymptotic expansions. Using this method, we prove polynomial upper bounds on the decay of correlations for smooth observables that match the optimal rates originally obtained by Ratner for the standard horocycle flow. Furthermore, in the presence of a spectral gap below , we establish exact polynomial asymptotics, mirroring the classical behavior of the horocycle flow.

Paper Structure

This paper contains 14 sections, 20 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Time-changes
  5. Ergodic integrals for time-changes
  6. Push-forward of geodesic arcs
  7. A refined mixing-via-shearing argument
  8. Variations on Forni and Ulcigrai's result
  9. The key lemma
  10. Proof of \ref{['thm:main']}
  11. Proofs of \ref{['thm:main_2']} and \ref{['prop:correlations_with_small_integral']}
  12. Proof of \ref{['thm:main_2']}
  13. Proof of \ref{['prop:correlations_with_small_integral']}
  14. Correlations of coboundaries

Key Result

Theorem 1

Let $\alpha \in W^6(M)$ be a positive function, and let $\{\mathsf{h}^{\alpha}_t\}_{t\in{ \mathbb R}}$ be the associated time-change of the horocycle flow on $M$. There exists a constant $C_{\alpha}>0$ such that, for any zero-average function $f\in W^6(M)$, for any $\ell \in W^1(M)$, and for any $t\

Theorems & Definitions (33)

  • Theorem 1: FU
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 23 more