Hypercyclicity of Shifts on Weighted ${\mathbf L}^p$ Spaces of Directed Trees

TL;DR

This paper investigates the conditions under which forward and backward shift operators on weighted L^p spaces over directed trees are hypercyclic, revealing that nontrivial trees can support hypercyclic backward shifts while only trivial trees support forward shifts.

Contribution

It provides necessary and sufficient conditions for hypercyclicity of backward shifts on weighted L^p spaces of directed trees, extending classical results to new tree structures.

Findings

Forward shifts are hypercyclic only on trivial trees.

Backward shifts can be hypercyclic on nontrivial trees.

Conditions for hypercyclicity coincide on unweighted rooted trees.

Abstract

In this paper, we study the hypercyclicity of forward and backward shifts on weighted $L^p$ spaces of a directed tree. In the forward case, only the trivial trees may support hypercyclic shifts, in which case the classical results of Salas apply. For the backward case, nontrivial trees may support hypercyclic shifts. We obtain necessary conditions and sufficient conditions for hypercyclicity of the backward shift and, in the case of a rooted tree on an unweighted space, we show that these conditions coincide.