  Let $X$ be a complete, simply connected harmonic manifold with sectional
curvatures $K$ satisfying $K \leq -1$. In \cite{biswas6}, a Fourier transform
was defined for functions on $X$, and a Fourier inversion formula and
Plancherel theorem were proved. We use the Fourier transform to investigate the
dynamics on $L^p(X)$ for $p > 2$ of certain bounded linear operators $T :
L^p(X) \to L^p(X)$ which we call "$L^p$-multipliers" in accordance with
standard terminology. These operators are required to preserve the subspace of
$L^p$ radial functions. A notion of convolution with radial functions was
defined in \cite{biswas6}, and these operators are also required to be
compatible with convolution in the sense that $$ T\phi * \psi = \phi * T\psi $$
for all radial $C^{\infty}_c$-functions $\phi, \psi$. They are also required to
be compatible with translation of radial functions. Examples of
$L^p$-multipliers are given by the operator of convolution with an $L^1$ radial
function, or more generally convolution with a finite radial measure. In
particular elements of the heat semigroup $e^{t\Delta}$ act as multipliers.
Given $2 < p < \infty$, we show that for any $L^p$-multiplier $T$ which is not
a scalar multiple of the identity, there is an open set of values of $\nu \in
\mathbb{C}$ for which the operator $\frac{1}{\nu} T$ is chaotic on $L^p(X)$ in
the sense of Devaney, i.e. topologically transitive and with periodic points
dense. Moreover such operators are topologically mixing. We also show that
there is a constant $c_p > 0$ such that for any $c \in \mathbb{C}$ with $\Re c
> c_p$, the action of the shifted heat semigroup $e^{ct} e^{t\Delta}$ on
$L^p(X)$ is chaotic. These results generalize the corresponding results for
rank one symmetric spaces of noncompact type and negatively curved harmonic
$NA$ groups (or Damek-Ricci spaces).
