A note on weak Banach mean equicoontinuity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917570102165504 |
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| author | Yang, Zhongxuan Huang, Xiaojun |
| author_facet | Yang, Zhongxuan Huang, Xiaojun |
| contents | Consider a topological dynamical system $(X, T)$ endowed with the metric $d$. We introduce a novel function as $\overline{BF}(x, y) = \limsup_{n-m \rightarrow +\infty} \inf_{σ\in S_{n,m}} \frac{1}{n-m} \sum_{k=m}^{n-1} d\left(T^{k} x, T^{σ(k)} y\right)$, where the permutation group $S_{n,m}$ is utilized. It is demonstrated that $BF(x, y)$ exists when $x, y \in X$ are uniformly generic points. Leveraging this function, we introduce the concept of weak Banach mean equicontinuity and establish that the dynamical system $(X, T)$ exhibits weak Banach mean equicontinuity if and only if the uniform time averages $f_B^{*}(x) = \lim_{n-m \rightarrow +\infty} \frac{1}{n-m} \sum_{k=m}^{n-1} f\left(T^{k} x\right)$ are continuous for all $f \in C(X)$. Finally, we demonstrate that in the case of a transitive system, the equivalence between weak Banach mean equicontinuity and weak mean equicontinuity is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09744 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on weak Banach mean equicoontinuity Yang, Zhongxuan Huang, Xiaojun Dynamical Systems Consider a topological dynamical system $(X, T)$ endowed with the metric $d$. We introduce a novel function as $\overline{BF}(x, y) = \limsup_{n-m \rightarrow +\infty} \inf_{σ\in S_{n,m}} \frac{1}{n-m} \sum_{k=m}^{n-1} d\left(T^{k} x, T^{σ(k)} y\right)$, where the permutation group $S_{n,m}$ is utilized. It is demonstrated that $BF(x, y)$ exists when $x, y \in X$ are uniformly generic points. Leveraging this function, we introduce the concept of weak Banach mean equicontinuity and establish that the dynamical system $(X, T)$ exhibits weak Banach mean equicontinuity if and only if the uniform time averages $f_B^{*}(x) = \lim_{n-m \rightarrow +\infty} \frac{1}{n-m} \sum_{k=m}^{n-1} f\left(T^{k} x\right)$ are continuous for all $f \in C(X)$. Finally, we demonstrate that in the case of a transitive system, the equivalence between weak Banach mean equicontinuity and weak mean equicontinuity is established. |
| title | A note on weak Banach mean equicoontinuity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2401.09744 |