Dynamics of Fourier Multipliers on Riemannian Symmetric Spaces of Noncompact Type

Let X be a Riemannian symmetric space of noncompact type and T be a linear translation-invariant operator which is bounded on Lᵖ(X) . We shall show that if T is not a constant multiple of identity then there exist complex constants z such that zT is chaotic on Lᵖ(X) when p is in the sharp range 2<p<∞ . This vastly generalizes the result that dynamics of the (perturbed) heat semigroup is chaotic on X proved in Ji and Weber (Ergodic Theory Dynam. Systems 30 (2010), 457–468) and Pramanik and Sarkar (J. Funct. Anal. 266 (2014), 2867–2909).