Flows in fibers

Abstract

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