Ergodic Theory and Dynamical Systems

Research Article

Dynamics of the heat semigroup on symmetric spaces

LIZHEN JIa1 and ANDREAS WEBERa2

a1 Department of Mathematics, University of Michigan, 1834 East Hall, Ann Arbor, MI 48109-1043, USA (email: lji@umich.edu)

a2 Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstrasse 2, 76128 Karlsruhe, Germany (email: andreas.weber@math.uni-karlsruhe.de)

Abstract

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)