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

Pratyoosh Kumar; Sumit Kumar Rano

doi:10.5802/crmath.382

Analyse harmonique, Systèmes dynamiques

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

Pratyoosh Kumar ¹ ; Sumit Kumar Rano ¹

¹ Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1-13.

Résumé

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 $T (t) = e^{t Ψ (ℒ)}$ to be chaotic on $L^{p}$ -spaces. We also study the chaotic dynamics of the semigroup $T (t) = e^{t (a ℒ + b)}$ 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.

Reçu le : 2022-02-02
Révisé le : 2022-05-18
Accepté le : 2022-05-20
Publié le : 2023-01-12

DOI : 10.5802/crmath.382

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

Affiliations des auteurs :

Pratyoosh Kumar ¹ ; Sumit Kumar Rano ¹

¹ Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Pratyoosh Kumar and Sumit Kumar Rano},
     title = {Dynamics of semigroups generated by analytic functions of the {Laplacian} on {Homogeneous} {Trees}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--13},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.382},
     language = {en},
}

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Pratyoosh Kumar; Sumit Kumar Rano. Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1-13. doi : 10.5802/crmath.382. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.382/

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