Flows in fibers

Abstract:

Let ${H^\infty }(\Delta )$ be the algebra of all bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak {M}({H^\infty }(\Delta ))$ be the maximal ideal space of ${H^\infty }(\Delta )$. Using a flow, we represent a reasonable portion of a fiber in $\mathfrak {M}({H^\infty }(\Delta ))$. 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 $\mathfrak {M}({H^\infty }(\Delta ))$ may contain a homeomorphic copy of $\mathfrak {M}({H^\infty }(\Delta ))$. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.