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

We study here the action of subgroups of PSL(2, 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 Lionville 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 Z-action generated by hyperbolic or parabolic elements of PSL(2, R) is chaotic.