"Dynamics of Lp multipliers on harmonic manifolds" by Kingshook Biswas and Rudra P. Sarkar
 

Dynamics of Lp multipliers on harmonic manifolds

Article Type

Research Article

Publication Title

Electronic Research Archive

Abstract

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).

First Page

3042

Last Page

3057

DOI

10.3934/era.2022154

Publication Date

1-1-2022

Comments

Open Access, Gold, Green

This document is currently not available here.

Share

COinS