Invariant measures for quasiperiodically forced circle homeomorphisms

We study quasiperiodically forced circle homeomorphisms and derive a basic classification with respect to the invariant ergodic measures for such systems: Either there exists an invariant graph and every invariant ergodic measure is associated to some invariant graph, or the system is uniquely ergodic. This immediately verifies an observation which is well-known from numerical studies, namely that the existence of (pointwise) non-zero Lyapunov exponents implies the existence of invariant graphs. Further more we prove a statement about the dynamical behaviour of typical orbits, which does not depend on the differentiability of the system and its Lyapunov exponents.