On the chaotic behavior of the Dunkl heat semigroup on 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 , (ϕiρ(x)) 2 dx) where ϕiρ 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.2010 Mathematics Subject Classification. Primary: 43A85; Secondary: 22E30.

Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

“…This difference can also be seen in Theorems A and B of this paper. Similar results concerning the chaotic behaviour of the L p -heat semigroup and the weighted L p -Dunkl heat semigroup are also known on harmonic N A groups and Euclidean spaces, respectively (see [1,Theorem 1.3], and [15,Theorem A]). Another motivation to study this class of semigroups is the work of deLaubenfels and Emamirad [11].…”

“…Theorem A. Let 2 < p < ∞ and T (t ) = e t Ψ(L ) be a semigroup on L p (X) as defined in (1). Then the following statements are equivalent.…”

“…Let Ψ be a nonconstant complex holomorphic function defined on a connected open set containing σ p (L ). Then by the usual Riesz functional calculus (see [14,Definition 10.26 at page 261]), it follows that the semigroup T (t ) = e t Ψ(L ) , where t ≥ 0, (1) consists of bounded linear operators on L p (X) for every p ∈ [1, ∞]. Moreover, the semigroup T (t ) is strongly continuous, that is, T (t ) f − f L p (X) → 0 as t → 0, for every f ∈ L p (X).…”

Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees

“…Hence by Theorem 3.1 (b) it follows that P σ p (T (t)) = ∅ for all t > 0. This show that only the zero function is a periodic point, that is, T (t) has no non-trivial periodic point in L p (X) for any p ∈[1,2].Part (…”

“…Dunkl heat semigroup) is also known for harmonic N A groups (resp. Euclidean spaces) (see [1] and [15]). In [13], the authors proved the following result: Theorem 1.2.…”