ℱ-Transitivity of Translation Semigroups on Directed Metric Trees
Abstract
We study the ℱ-transitivity and topological ℱ-recurrence of left translation semigroups on weighted Lp-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 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 ℱ-transitivity and topological 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 ℱ-transitivity in the rootless setting.
Related
- Approximate Transitivity of Young Translation on Rough Paths
- Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees
- Wold-type decompositions for norm-increasing m-concave weighted shifts on directed trees
- Gaussian Processes on Directed Metric Graphs
- Loop soup percolation and boundary dimension on metric trees
BibTeX
@misc{chen2026ℱtransitivity,
title = {{$\mathcalF$-Transitivity of Translation Semigroups on Directed Metric Trees}},
author = {Xiang Chen and Li Zhang and Zehua Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02342},
eprint = {2609.02342},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02342},
}
RIS
TY - UNPB
TI - ℱ-Transitivity of Translation Semigroups on Directed Metric Trees
AU - Xiang Chen
AU - Li Zhang
AU - Zehua Zhou
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02342
UR - https://arxiv.org/abs/2609.02342
AB - We study the ℱ-transitivity and topological 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 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 ℱ-transitivity and topological 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 ℱ-transitivity in the rootless setting.
ER -