Pointwise convergence of some continuous-time polynomial ergodic averages

2023/10/25 by Wen Huang, Song Shao, Huang, Wen +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2310.16780

openalex publication_date 2023/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ ℝ, Q∈ ℝ[t] with deg Q≥ 2. Let (X,X,μ, (T^t)_{t∈ ℝ}) and (X,X,μ, (S^t)_{t∈ ℝ}) be two measurable flows. Then for any f₁, f₂, g∈ L^∞(μ), the limit lim_M→∞(1)/(M)∫₀^Mf₁(T^tx)f₂(T^atx)g(S^Q(t)x)dt exists for μ-a.e. x∈ X. In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.

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