  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.
