Flows in fibers

Let be the algebra of all bounded analytic functions on the open unit disc Δ, and let be the maximal ideal space of Using a flow, we represent a reasonable portion of a fiber in This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in may contain a homeomorphic copy of Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.