Ergodic Theory and Dynamical Systems

Research Article

Dynamics of the heat semigroup on symmetric spaces


a1 Department of Mathematics, University of Michigan, 1834 East Hall, Ann Arbor, MI 48109-1043, USA (email:

a2 Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstrasse 2, 76128 Karlsruhe, Germany (email:


The aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

(Received August 18 2008)

(Revised February 12 2009)