Dynamics of the heat semigroup on symmetric spaces

Abstract

The aim of this paper is to show that the dynamics of L p heat semigroups (p > 2) on a symmetric space of non-compact type is very different from the dynamics of the L p heat semigroups if 1 < p ≤ 2. To see this, it is shown that certain shifts of the L p heat semigroups have a chaotic behavior if p > 2, and that such a behavior is not possible in the cases 1 < p ≤ 2. These results are compared with the corresponding situation for euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

“…by performing the Laplace transform at the both sides of the composition property (31). By the injectivity of the operator − + ∑ ∈ − for = 0 , we obtain that = 0 and the claimed assertion follows.…”

“…For the sake of illustration, we will consider only the situation of [12, Example 3.3.12(ii)]; see also Ji and Weber [31]. Suppose that is a symmetric space of noncompact type and rank one, > 2, and the parabolic domain and the positive real number possess the same meaning as in [31]. Suppose, further, that ( ) = ∑ =0 , ∈ C is a nonconstant complex polynomial with > 0, = 2, 0 < < 2, …”

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

“…The study of chaotic dynamics of the heat semigroup on Riemannian symmetric spaces of noncompact type, which started with the work of Ji and Weber [10] has been completed recently by Pramanik and Sarkar [13] (see also Sarkar [17]). As they have remarked, the chaotic behavior of the heat semigroups on L p spaces seems to be a non-Euclidean phenomenon.…”

Dynamics of multivalued linear operators

“…It can be easily checked that the sets of periodic points of these operators are dense in Q X : Example 3.12. Let p > 2 and let X be a symmetric space of non-compact type of rank one, let P p be the parabolic domain defined in the proof of [29,Th. 3.1], and let c p > 0 be the apex of…”