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

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 $L^p({\mathbb{R}}^n,({\phi }_{i\rho }(x){)}^{2}dx)$ where ${\phi }_{i\rho }$ 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.