Comptes Rendus
Analyse harmonique, Systèmes dynamiques
Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1-13.

Let Ψ be a non-constant complex-valued analytic function defined on a connected, open set containing the L p -spectrum of the Laplacian on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup T(t)=e tΨ() to be chaotic on L p -spaces. We also study the chaotic dynamics of the semigroup T(t)=e t(a+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 :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.382
Classification : 39A12, 43A85, 47D06, 47A16, 39A33

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
@article{CRMATH_2023__361_G1_1_0,
     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},
}
TY  - JOUR
AU  - Pratyoosh Kumar
AU  - Sumit Kumar Rano
TI  - Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1
EP  - 13
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.382
LA  - en
ID  - CRMATH_2023__361_G1_1_0
ER  - 
%0 Journal Article
%A Pratyoosh Kumar
%A Sumit Kumar Rano
%T Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees
%J Comptes Rendus. Mathématique
%D 2023
%P 1-13
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.382
%G en
%F CRMATH_2023__361_G1_1_0
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/

[1] Pradeep Boggarapu; Sundaram Thangavelu On the chaotic behavior of the Dunkl heat semigroup on weighted L p spaces, Isr. J. Math., Volume 217 (2017) no. 1, pp. 57-92 | DOI | MR | Zbl

[2] Joel M. Cohen; Mauro Pagliacci; Massimo A. Picardello Universal properties of the isotropic Laplace operator on homogeneous trees, Adv. Math., Volume 401 (2022), 108311, 9 pages | DOI | MR | Zbl

[3] Michael Cowling; Stefano Meda; Alberto G. Setti Estimates for functions of the Laplace operator on homogeneous trees, Trans. Am. Math. Soc., Volume 352 (2000) no. 9, pp. 4271-4293 | DOI | MR | Zbl

[4] Wolfgang Desch; Wilhelm Schappacher; Glenn F. Webb Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dyn. Syst., Volume 17 (1997) no. 4, pp. 793-819 | DOI | MR | Zbl

[5] Robert L. Devaney An introduction to chaotic dynamical systems, Addition-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Group, 1989 | Zbl

[6] Alessandro Figà-Talamanca; Massimo A. Picardello Spherical functions and harmonic analysis on free groups, J. Funct. Anal., Volume 47 (1982) no. 3, pp. 281-304 | DOI | MR | Zbl

[7] Alessandro Figà-Talamanca; Massimo A. Picardello Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, 87, Marcel Dekker, 1983 | Zbl

[8] Loukas Grafakos Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, 2014 | Zbl

[9] Lizhen Ji; Andreas Weber Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dyn. Syst., Volume 30 (2010) no. 2, pp. 457-468 | MR | Zbl

[10] Pratyoosh Kumar; Sumit K. Rano A characterization of weak L p -eigenfunctions of the Laplacian on Homogeneous trees, Ann. Mat. Pura Appl., Volume 200 (2021) no. 2, pp. 721-736 | DOI | MR | Zbl

[11] Ralph J. de Laubenfels; Hassan Emamirad Chaos for functions of discrete and continuous weighted shift operators, Ergodic Theory Dyn. Syst., Volume 21 (2001) no. 5, pp. 1411-1427 | MR | Zbl

[12] Frank W. J. Olver Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press Inc., 1974 | Zbl

[13] Malabika Pramanik; Rudra P. Sarkar Chaotic dynamics of the heat semigroup on Riemannian symmetric spaces, J. Funct. Anal., Volume 266 (2014) no. 5, pp. 2867-2909 | DOI | MR | Zbl

[14] Walter Rudin Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., 1973 | Zbl

[15] Rudra P. Sarkar Chaotic dynamics of the heat semigroup on the Damek-Ricci spaces, Isr. J. Math., Volume 198 (2013) no. 1, pp. 487-508 | DOI | MR | Zbl

[16] Alberto G. Setti L p and operator norm estimates for the complex time heat operator on homogeneous trees, Trans. Am. Math. Soc., Volume 350 (1998) no. 2, pp. 743-768 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique