Let be a non-constant complex-valued analytic function defined on a connected, open set containing the -spectrum of the Laplacian on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup to be chaotic on -spaces. We also study the chaotic dynamics of the semigroup separately and obtain a sharp range of for which is chaotic on -spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.
Revised:
Accepted:
Published online:
Pratyoosh Kumar 1; Sumit Kumar Rano 1
@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 -
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] On the chaotic behavior of the Dunkl heat semigroup on weighted spaces, Isr. J. Math., Volume 217 (2017) no. 1, pp. 57-92 | DOI | MR | Zbl
[2] Universal properties of the isotropic Laplace operator on homogeneous trees, Adv. Math., Volume 401 (2022), 108311, 9 pages | DOI | MR | Zbl
[3] 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] Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dyn. Syst., Volume 17 (1997) no. 4, pp. 793-819 | DOI | MR | Zbl
[5] An introduction to chaotic dynamical systems, Addition-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Group, 1989 | Zbl
[6] Spherical functions and harmonic analysis on free groups, J. Funct. Anal., Volume 47 (1982) no. 3, pp. 281-304 | DOI | MR | Zbl
[7] Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, 87, Marcel Dekker, 1983 | Zbl
[8] Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, 2014 | Zbl
[9] Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dyn. Syst., Volume 30 (2010) no. 2, pp. 457-468 | MR | Zbl
[10] A characterization of weak eigenfunctions of the Laplacian on Homogeneous trees, Ann. Mat. Pura Appl., Volume 200 (2021) no. 2, pp. 721-736 | DOI | MR | Zbl
[11] 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] Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press Inc., 1974 | Zbl
[13] 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] Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., 1973 | Zbl
[15] 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] 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
Cited by Sources:
Comments - Policy