$\mathcal{F}$-Transitivity of Translation Semigroups on Directed Metric Trees

Abstract

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.