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Pointwise convergence of some continuous-time polynomial ergodic averages

Wen Huang, Song Shao, Rongzhong Xiao

TL;DR

The paper establishes pointwise convergence results for continuous-time polynomial ergodic averages involving two flows and a polynomial time-change of degree at least two. By leveraging topological models of measurable flows and maximal inequalities, the authors reduce continuous-time averages to discrete-time ergodic averages, then apply known pointwise convergence theorems to obtain almost-everywhere limits. The main contributions include Theorem A (existence of a pointwise limit), a refinement in Theorem B that expresses the limit via conditional expectations against the invariant factors, a specialized product-form result in Theorem C for linear degrees, and a multi-flow extension in Theorem D with applications to geodesic and horocycle flows (Corollary E). The work broadens the reach of continuous-time analogues of Furstenberg–Bergelson–Leibman conjectures, providing new tools for analyzing multi-parameter ergodic averages in smooth dynamical settings and highlighting the role of zero-entropy-like reductions and nilpotent-like structures for pointwise convergence. Overall, the results advance the understanding of pointwise convergence in noncommutative, continuous-time ergodic theory with concrete geometric applications.

Abstract

In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in \mathbb{R}$, $Q\in \mathbb{R}[t]$ with $\text{deg}\ Q\ge 2$. Let $(X,\mathcal{X},μ, (T^{t})_{t\in \mathbb{R}})$ and $(X,\mathcal{X},μ, (S^{t})_{t\in \mathbb{R}})$ be two measurable flows. Then for any $f_1, f_2, g\in L^{\infty}(μ)$, the limit \begin{equation*} \lim\limits_{M\to\infty}\frac{1}{M}\int_{0}^{M}f_1(T^{t}x)f_2(T^{at}x)g(S^{Q(t)}x)dt \end{equation*} exists for $μ$-a.e. $x\in X$. In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.

Pointwise convergence of some continuous-time polynomial ergodic averages

TL;DR

The paper establishes pointwise convergence results for continuous-time polynomial ergodic averages involving two flows and a polynomial time-change of degree at least two. By leveraging topological models of measurable flows and maximal inequalities, the authors reduce continuous-time averages to discrete-time ergodic averages, then apply known pointwise convergence theorems to obtain almost-everywhere limits. The main contributions include Theorem A (existence of a pointwise limit), a refinement in Theorem B that expresses the limit via conditional expectations against the invariant factors, a specialized product-form result in Theorem C for linear degrees, and a multi-flow extension in Theorem D with applications to geodesic and horocycle flows (Corollary E). The work broadens the reach of continuous-time analogues of Furstenberg–Bergelson–Leibman conjectures, providing new tools for analyzing multi-parameter ergodic averages in smooth dynamical settings and highlighting the role of zero-entropy-like reductions and nilpotent-like structures for pointwise convergence. Overall, the results advance the understanding of pointwise convergence in noncommutative, continuous-time ergodic theory with concrete geometric applications.

Abstract

In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let , with . Let and be two measurable flows. Then for any , the limit \begin{equation*} \lim\limits_{M\to\infty}\frac{1}{M}\int_{0}^{M}f_1(T^{t}x)f_2(T^{at}x)g(S^{Q(t)}x)dt \end{equation*} exists for -a.e. . In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.
Paper Structure (23 sections, 14 theorems, 166 equations)

This paper contains 23 sections, 14 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Furstenberg-Bergelson-Leibman conjecture
  3. Continuous-time analogues of Furstenberg-Bergelson-Leibman conjecture
  4. Ergodic theorems without commutativity
  5. Main results
  6. A brief overview of the proofs
  7. Preliminaries
  8. Notations
  9. Conditional expectation and disintegration of a measure
  10. Topological models of measurable flows
  11. Maximal ergodic theorems
  12. Pointwise ergodic theorems for discrete-time actions
  13. Pointwise ergodic theorems for measurable flows
  14. Proofs of Theorem B and Theorem C
  15. Two lemmas
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $T,S$ be invertible measure preserving transformations acting on Lebesgue probability space $(X,\mathcal{X},\mu)$ such that $(X,\mathcal{X},\mu,T)$ has zero entropy. Let $P\in \mathbb Z[n]$ with $\deg P \ge 2$. Then for any $f,g\in L^\infty(\mu)$, the limit exists in $L^2(\mu)$.

Theorems & Definitions (24)

  • Remark 1.2
  • Theorem 1.3: HN
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Theorem 2.2: V
  • Theorem 2.3: KLMP
  • Theorem 2.4: IAMMS
  • Theorem 2.5: B2
  • ...and 14 more