Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees
Let $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaoti...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.382/ |
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| Summary: | Let $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaotic on $L^{p}$-spaces. We also study the chaotic dynamics of the semigroup $T(t)=e^{t(a\mathcal{L}+b)}$ 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. |
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| ISSN: | 1778-3569 |