On the chaotic behavior of the Dunkl heat semigroup on weighted $ L^p $ spaces

TL;DR

This paper investigates the chaotic dynamics of the Dunkl heat semigroup on weighted L^p spaces, extending known results from classical Laplacian cases to Dunkl operators and revealing similar behaviors.

Contribution

It provides a comprehensive analysis of chaos in Dunkl heat semigroups on weighted L^p spaces, paralleling results known for Laplace-Beltrami operators.

Findings

Chaotic behavior characterized for Dunkl heat semigroup.

Similarity to Laplace-Beltrami operator on symmetric spaces.

Complete description on specific weighted L^p spaces.

Abstract

In this paper we study the chaotic behaviour of the heat semigroup generated by the Dunkl-Laplacian on weighted $ L^p$ spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces $ L^p(\R^n, (\varphi_{i\rho}(x))^2 dx) $ where $ \varphi_{i\rho} $ is the Euclidean spherical function. The behaviour is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.