  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
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 Z-action generated by hyperbolic or parabolic
elements of PSL(2,R) is chaotic.
