Dynamics of $L^p$ multipliers on harmonic manifolds

TL;DR

This paper studies the dynamics of certain linear operators called $L^p$-multipliers on harmonic manifolds, showing that many such operators exhibit chaotic behavior under specific conditions, extending known results from symmetric spaces.

Contribution

It introduces a framework for analyzing $L^p$-multipliers on harmonic manifolds and proves their chaotic dynamics, generalizing previous results from symmetric spaces and Damek-Ricci spaces.

Findings

$L^p$-multipliers can be chaotic on harmonic manifolds for certain parameters.

Shifted heat semigroup operators are chaotic for large enough real parts of the shift.

Results extend known chaos phenomena from symmetric spaces to harmonic manifolds.

Abstract

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).