@article{Biswas2022, 
author = {Kingshook Biswas and Rudra P. Sarkar},
title = {Dynamics of  Lp multipliers on harmonic manifolds},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {8},
pages = {3042-3057},
keywords = {Fourier transform, heat semigroup, Lp multipliers, harmonic manifolds, Devaney chaotic},
url = {https://www.sciopen.com/article/10.3934/era.2022154},
doi = {10.3934/era.2022154},
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  etΔ 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  1νT 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  ectetΔ 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).}
}