Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

Abstract

Let Ψ be a non-constant complex-valued analytic function defined on a connected, open set containing the L p -spectrum of the Laplacian ℒ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup T(t)=e tΨ(ℒ) to be chaotic on L p -spaces. We also study the chaotic dynamics of the semigroup T(t)=e t(aℒ+b) separately and obtain a sharp range of b for which T(t) is chaotic on L p -spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.