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\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 $ \operatorname{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 harmonic $ NA $ groups (or Damek-Ricci spaces).