Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T05:50:55.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics on the space of harmonic functions and the foliated Liouville problem

Published online by Cambridge University Press:  02 February 2005

R. FERES  and
A. ZEGHIB
R. FERES
Affiliation:
Department of Mathematics, 1146 Washington University, St. Louis, MO 63130, USA (e-mail: [email protected])
A. ZEGHIB
Affiliation:
UMPA, École Normale Supérieure de Lyon, 69364 Lyon CEDEX 07, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We study here the action of subgroups of PSL$(2,\mathbb{R})$ on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem. Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on the manifold, is the function leafwise constant? We give a number of positive results and also show a general class of examples for which the Liouville property does not hold. The connection between the Liouville property and the dynamics on the space of harmonic functions as well as general properties of this dynamical system are explored. It is shown among other properties that the $\mathbb{Z}$-action generated by hyperbolic or parabolic elements of PSL$(2,\mathbb{R})$ is chaotic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 2 , April 2005 , pp. 503 - 516
Copyright
2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)