ON SOME HOLOMORPHIC DYNAMICAL SYSTEMS
Quarterly Journal of Mathematics
Let D be a domain in C N 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 where B(D) denotes the Borel σ-algebra of D. Adopting the terminology introduced in , we will say that ϕ * is finite if it has a fixed point among probability measures. Let L 2 H(D) be the Hilbert space of all square
... of all square Lebesgue integrable holomorphic functions on D. The purpose of this paper is to exhibit various relations between ϕ, ϕ * , and U ϕ . A fundamental result is that U ϕ has either pure point spectrum or purely continuous spectrum, the first case occurring exactly when ϕ * is finite. We prove that any of the following two conditions ensures the finiteness of ϕ * : 1 o the existence of a ϕ-invariant probability measure absolutely continuous with respect to Lebesgue measure; 2 o the existence of a relatively compact orbit of ϕ. Of course, the first condition is also necessary. We show the necessity of a stronger version of the second condition (embeddability of ϕ in a compact transformation group) provided some mild restrictions on D are imposed. Assuming some hypotheses on D, we prove also that if a point in D is wandering, then U ϕ has purely absolutely continuous spectrum, and, conversely, if U ϕ has a non-zero absolutely continuous component in the spectrum, then all points in D are wandering. In particular, the spectrum of U ϕ is either pure point, or purely absolutely continuous, or purely singular continuous. We show that if D is in a class of domains containing among others all bounded analytic polyhedra, then the spectrum of U ϕ cannot be purely singular continuous.