AI 中文总结
该研究针对有向度量树上加权$L^p$空间的左平移半群，建立了$\boldsymbol{\textit{F}}$-传递性与拓扑$\boldsymbol{\textit{F}}$-回归性的等价条件，明确了有根树与无根树的不同判定准则，揭示了分支对动力学的影响。
AI 中文摘要
我们研究有向度量树上加权$L^p$空间上左平移半群的$\boldsymbol{\textit{F}}$-传递性与拓扑$\boldsymbol{\textit{F}}$-回归性。受Mangino和Vargas-Moreno近期关于这些半群的超循环性与弱混合性工作的启发，我们研究$\boldsymbol{\textit{F}}$-传递性这一不同问题，其中要求完整的回归时间集属于给定的有限不变Furstenberg族。在权重为$p$-容许的假设下，我们分别针对有根树与无根树得到了充分必要积分条件，并建立了$\boldsymbol{\textit{F}}$-传递性与拓扑$\boldsymbol{\textit{F}}$-回归性之间的等价关系。对于有根树，判定准则仅依赖于后代边的权重；对于无根树，额外的祖先项与可测消去构造反映了树的反向几何结构。我们的方法基于可测加权极小化与边参数区间子集上的定量估计。关于齐次有根树和具有自由左端的无根树的例子表明，分支可能决定动力学，但在无根情形下仅前向分支无法保证$\boldsymbol{\textit{F}}$-传递性。
英文摘要
We study the $\mathcal F$-transitivity and topological $\mathcal F$-recurrence of left translation semigroups on weighted $L^p$-spaces over directed metric trees. Motivated by the recent work of Mangino and Vargas-Moreno on hypercyclicity and weak mixing for these semigroups, we investigate the different problem of $\mathcal F$-transitivity, where the entire return-time set is required to belong to a prescribed finitely invariant Furstenberg family. Assuming that the weight is $p$-admissible, we obtain necessary and sufficient integral conditions for both rooted and rootless trees, and establish the equivalence between $\mathcal F$-transitivity and topological $\mathcal F$-recurrence. In the rooted case, the criteria depend only on the weights along descendant edges. In the rootless case, an additional ancestor term and a measurable cancellation construction reflect the backward geometry of the tree. Our approach is based on measurable weighted minimization and quantitative estimates over subsets of the edge parameter interval. Examples on homogeneous rooted trees and rootless trees with a free left end show that branching may determine the dynamics, but forward branching alone does not guarantee $\mathcal F$-transitivity in the rootless setting.
Comments27 pages, no figures
