Chaotic Dynamics of the heat semigroup on the Damek-Ricci spaces

TL;DR

This paper proves that certain heat semigroups on Damek-Ricci spaces exhibit chaotic behavior on Lorentz and weak L^p spaces, extending previous results on symmetric spaces.

Contribution

It establishes chaos of the heat semigroup on Damek-Ricci spaces for specific perturbations and p-ranges, generalizing prior work on symmetric spaces.

Findings

Heat semigroup is chaotic on Lorentz spaces for 2<p<0, 1\u2264 q<0.

The perturbation size and p-range are shown to be sharp.

Generalizes previous results from symmetric to Damek-Ricci spaces.

Abstract

The Damek-Ricci spaces are solvable Lie groups and noncompact harmonic manifolds. The rank one Riemannian symmetric spaces of noncompact type sits inside it as a thin subclass. In this note we establish that for any Damek-Ricci space $S$, the heat semigroup generated by certain perturbation of the Laplace-Beltrami operator is {\em chaotic} on the Lorentz spaces $L^{p,q}(S)$, $2<p<\infty, 1\le q<\infty$ and subspace-chaotic on the weak $L^p$-spaces. We show that both the amount of perturbation and the range of $p$ are sharp. This generalizes a result in \cite{J-W} which proves that under identical conditions, the heat semigroup mentioned above is {\em subspace-chaotic} on the $L^p$-spaces of the symmetric spaces.