Dynamics of Lp multipliers on harmonic manifolds

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Research Article

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Electronic Research Archive


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 Lp (X) for p > 2 of certain bounded linear operators T: Lp (X) Lp (X) which we call ”Lp-multipliers” in accordance with standard terminology. Examples of Lp-multipliers are given by the operator of convolution with an L1 radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup et∆ act as multipliers. Given 2 < p >, we show that for any Lp-multiplier T which is not a scalar multiple of the identity, there is an open set of values of ν C for which the operator 1 T is chaotic on Lp (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 cp > 0 such that for any c C with Re c > cp, the action of the shifted heat semigroup ectet∆ on Lp (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).

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Open Access, Gold, Green

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