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Effective Equidistribution for Contact Anosov flows in Dimension Three

Asaf Katz, Thomas Aloysius O'Hare

TL;DR

This work proves effective equidistribution of Bowen packets for Anosov flows on three-dimensional manifolds, achieving exponential convergence for contact flows with $P(\Psi)>0$ when tested against $C^{1}$ observables. The core method combines a $C^{1}$-coding (via Bowen–Ratner refinement) with a two-variable zeta function framework, leveraging Dolgopyat-type bounds to establish a zero-free strip and deduce quantitative equidistribution, then transfers the symbolic estimates back to the original flow. For general Anosov flows, the authors obtain polynomial-rate equidistribution by using Hölder-coding and Pollicott–Sharp estimates, outlining a path from symbolic dynamics to dynamical averages. Overall, the paper links analytic properties of dynamical zeta functions and transfer operators to rigorous, rate-controlled equidistribution results, with concrete implications for geodesic flows and broader Anosov systems.

Abstract

We prove effective equidistribution theorems for (weighted) packets of closed periodic orbits for Anosov flows. In particular, for the case of contact Anosov flows on three-dimensional manifolds, we show that the Bowen packets equidistribute at an exponential rate.

Effective Equidistribution for Contact Anosov flows in Dimension Three

TL;DR

This work proves effective equidistribution of Bowen packets for Anosov flows on three-dimensional manifolds, achieving exponential convergence for contact flows with when tested against observables. The core method combines a -coding (via Bowen–Ratner refinement) with a two-variable zeta function framework, leveraging Dolgopyat-type bounds to establish a zero-free strip and deduce quantitative equidistribution, then transfers the symbolic estimates back to the original flow. For general Anosov flows, the authors obtain polynomial-rate equidistribution by using Hölder-coding and Pollicott–Sharp estimates, outlining a path from symbolic dynamics to dynamical averages. Overall, the paper links analytic properties of dynamical zeta functions and transfer operators to rigorous, rate-controlled equidistribution results, with concrete implications for geodesic flows and broader Anosov systems.

Abstract

We prove effective equidistribution theorems for (weighted) packets of closed periodic orbits for Anosov flows. In particular, for the case of contact Anosov flows on three-dimensional manifolds, we show that the Bowen packets equidistribute at an exponential rate.

Paper Structure

This paper contains 8 sections, 14 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Structure of the paper
  4. Encoding Anosov systems
  5. The Bowen-Ratner coding
  6. Coding flows with and without $C^{1}$-foliations
  7. Effective Equidistribution of Contact Anosov Flows via Zeta Functions
  8. General Anosov Flows

Key Result

Theorem 1.1

Let $g_t:M\rightarrow M$ be a contact Anosov flow on a three-dimensional manifold $M$, and let $\Psi\in C^1(M)$ be a potential with equilibrium state $\mu_\Psi$ and $P(\Psi)>0$, and let $\mu_{\Psi,T}$ be as in (eqn: Discrete Measure def). Then there exist constants $C,\delta>0$ such that for any $C^

Theorems & Definitions (23)

  • Theorem 1.1: Effective Equidistribution for contact flows
  • Remark 1
  • Remark 2
  • Theorem 1.2: Effective equidistribution for Anosov flows
  • Proposition 2.1: Bowen-Ratner coding, Bowen_1973Ratner_1973,Fisher_Hasselblatt_2019 Theorem 6.6.5
  • Lemma 2.2: $C^{1}$ lifting Lemma
  • proof
  • Lemma 2.3: Hölder lifting Lemma
  • proof
  • Proposition 3.1: Dolgopyat's estimate Dolgopyat_1998 Corollary $3$, see also Anantharaman_2000 Theorem $4.5$
  • ...and 13 more