Dynamics of $L^p$ multipliers on harmonic manifolds

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of nonpositive curvature, and in particular all known examples of non-compact harmonic manifolds except for the flat spaces. We use the Fourier transform from ^[1] 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. 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 $e^{t\mathrm{\Delta }}$ act as multipliers. Given $2<p<\mathrm{\infty }$, 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 $\mathrm{Re}c>c_p$, the action of the shifted heat semigroup $e^{ct}{e}^{t\mathrm{\Delta }}$ on $L^p(X)$ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and harmonic $NA$ groups (or Damek-Ricci spaces).