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

Kumar, Pratyoosh; Rano, Sumit Kumar

Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 1-13

Let $\Psi$ 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 $T\left(t\right)=e^{t\Psi \left(ℒ\right)}$ to be chaotic on $L^p$-spaces. We also study the chaotic dynamics of the semigroup $T\left(t\right)=e^{t(aℒ+b)}$ separately and obtain a sharp range of $b$ for which $T\left(t\right)$ is chaotic on $L^p$-spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.

Classification : 39A12, 43A85, 47D06, 47A16, 39A33

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