Dynamics of the heat semigroup on symmetric spaces

TL;DR

This paper investigates the differing dynamics of $L^p$ heat semigroups on symmetric spaces, revealing chaotic behavior for $p>2$ and contrasting it with the non-chaotic case for $p\

Contribution

It demonstrates that $L^p$ heat semigroups exhibit chaotic dynamics on non-compact symmetric spaces when $p>2$, a phenomenon not present for $p\leq 2$ or in other space types.

Findings

Chaotic behavior occurs for $p>2$ on non-compact symmetric spaces.

No chaotic behavior for $p\leq 2$ or in Euclidean and compact symmetric spaces.

Differences in dynamics highlight the influence of space geometry and $p$ on heat semigroup behavior.

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 $p\leq 2$. 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 $p\leq 2$. These results are compared with the corresponding situation for euclidean spaces and symmetric spaces of compact type where such a behavior is not possible.