Dynamics of the translation semigroup on directed metric trees

TL;DR

This paper studies the behavior of translation semigroups on weighted function spaces over directed metric trees, providing conditions for continuity and characterizing complex dynamical properties like hypercyclicity.

Contribution

It generalizes classical translation semigroup dynamics from Euclidean spaces to graph structures, offering new criteria based on weight decay for continuity and dynamical properties.

Findings

Conditions for strong continuity of the semigroup are established.

Hypercyclicity and weak mixing are characterized by weight decay along the tree.

Results extend classical $L^p$ translation dynamics to directed metric trees.

Abstract

The dynamics of the left translation semigroup $\{T_t\}_{t \geq 0}$ on weighted $L^p$ spaces over a directed metric tree $L(G)$ is investigated. Necessary and sufficient conditions on the weight family $\rho$ for the strong continuity of the semigroup are provided. Furthermore, hypercyclicity and weak mixing properties are characterized in terms of the asymptotic decay of $\rho$ along the tree structure. These results generalize classical $L^p$ translation semigroup dynamics to a graph setting.