Pointwise convergence of some continuous-time polynomial ergodic averages
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910825146482688 |
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| author | Huang, Wen Shao, Song Xiao, Rongzhong |
| author_facet | Huang, Wen Shao, Song Xiao, Rongzhong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_16780 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence of some continuous-time polynomial ergodic averages Huang, Wen Shao, Song Xiao, Rongzhong Dynamical Systems 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. |
| title | Pointwise convergence of some continuous-time polynomial ergodic averages |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2310.16780 |