On Some Holomorphic Dynamical Systems

Let D be a domain in C and φ a holomorphic automorphism of D. Let C be the measure class of the Lebesgue measure in D, i.e., the set of all positive regular Borel measures on D whose null sets coincide with the Lebesgue null sets. Let φ∗ be the automorphism of C given by (φ∗μ)(B) = μ(φ−1(B)) (μ ∈ C , B ∈ B(D)), where B(D) denotes the Borel σ-algebra of D. Adopting the terminology introduced in [4], we will say that φ∗ is finite if it has a fixed point among probability measures. Let LH(D) be the Hilbert space of all square Lebesgue integrable holomorphic functions on D. Suppose that LH(D) 6= {0}. Let Uφ be the unitary operator in LH(D) defined by Uφf = (f ◦ φ)Jφ (f ∈ LH(D)), where Jφ = det (∂φi ∂zj )