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- W4297663600 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 $$ Tphi * psi = phi * Tpsi $$ 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 $e^{tDelta}$ 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 $e^{ct} e^{tDelta}$ 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)." @default.
- W4297663600 created "2022-09-30" @default.
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- W4297663600 date "2018-05-28" @default.
- W4297663600 modified "2023-10-17" @default.
- W4297663600 title "Dynamics of $L^p$ multipliers on harmonic manifolds" @default.
- W4297663600 doi "https://doi.org/10.48550/arxiv.1805.10779" @default.
- W4297663600 hasPublicationYear "2018" @default.
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