Dynamics of $L^p$ multipliers on harmonic manifolds

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K\le -1$. 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 $T : L^p(X) \to L^p(X)$whichwecall\mathrm{"}$L^p$-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 $$T\varphi \ast \psi =\varphi \ast T\psi$$ for all radial $C_c^{\mathrm{\infty }}$-functions $\varphi ,\psi$. 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 $e^{t\mathrm{\Delta }}$ 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}$$forwhichtheoperator$\frac{1}{\nu} T$ischaoticon$L^p(X)$ 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$,theactionoftheshiftedheatsemigroup$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).