  Let $X$ be a Riemannian symmetric space of noncompact type and $T$ be a
linear translation-invariant operator which is bounded on $L^p(X)$. We shall
show that if $T$ is not a constant multiple of identity then there exist
complex constants $z$ such that $zT$ is chaotic on $L^p(X)$ when $p$ is in the
sharp range $2<p<\infty$. This vastly generalizes the result that dynamics of
the (perturbed) heat semigroup is chaotic on $X$ proved in [15, 17].
