Dynamics of the translation semigroup on directed metric trees

TL;DR

This work analyzes the left translation semigroup $(T_t)_{t\ge 0}$ on weighted $L^p$ spaces over directed metric trees $L(G)$, aiming to characterize strong continuity and chaotic dynamics. By constructing a path-based framework with the edge-incidence matrix $\mathcal{A}$ and edge-index sets $M_n(i)$, the authors derive sharp weight conditions, called $p$-admissible weights, that exactly determine when $T_t$ is a $C_0$-semigroup on $L^p_{\rho}(L(G))$. They provide complete criteria for hypercyclicity and weak mixing in terms of the asymptotic decay of weights along the tree, including explicit obstructions in trees with leaves and distinct rooted vs unrooted cases, along with discrete-time discretizations. The results generalize classical translation semigroup dynamics from the line to graph settings and illuminate how network topology and weight decay govern operator dynamics on graphs.

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 $ρ$ for the strong continuity of the semigroup are provided. Furthermore, hypercyclicity and weak mixing properties are characterized in terms of the asymptotic decay of $ρ$ along the tree structure. These results generalize classical $L^p$ translation semigroup dynamics to a graph setting.

Theorems & Definitions (18)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Definition 2.4