A Note on Random Holomorphic Iteration in Convex Domains

Bracci, Filippo

doi:10.1017/s0013091506001131

Cited by 1 publication

(1 citation statement)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A breakthrough occurred in 1988, when the first author (see [A1]) showed how to prove a Wolff-Denjoy theorem for holomorphic self-maps of smoothly bounded strongly convex domains in C n . The techniques introduced there turned out to be quite effective in other contexts too (see, e.g., [A5,AR,Br1,Br2]); but in particular they led to Wolff-Denjoy theorems in smooth strongly pseudoconvex domains and smooth domains of finite type (see, e.g., [A4,Hu,RZ,Br3]).…”

Section: Introductionmentioning

confidence: 99%

Wolff–Denjoy theorems in nonsmooth convex domains

Abate

Raissy

2013

Annali di Matematica

View full text Add to dashboard Cite

Abstract. We give a short proof of Wolff-Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff-Denjoy theorem for weakly convex domains, again without any smoothness assumption on the boundary. IntroductionStudying the dynamics of a holomorphic self-map f : ∆ → ∆ of the unit disk ∆ ⊂ C one is naturally led to consider two different cases. If f has a fixed point then Schwarz's lemma readily implies that either f is an elliptic automorphism, or the sequence {f k } of iterates of f converges (uniformly on compact sets) to the fixed point. The classical Wolff-Denjoy theorem ([W], [D]) says what happens when f has no fixed points:Theorem 0.1: (Wolff-Denjoy) Let f : ∆ → ∆ be a holomorphic self-map without fixed points. Then there exists a point τ ∈ ∂∆ such that the sequence {f k } of iterates of f converges (uniformly on compact sets) to the constant map τ .Since its discovery, a lot of work has been devoted to obtain similar statements in more general situations (surveys covering different aspects of this topic are [A3, RS, ES]). In one complex variable, there are results in multiply connected domains, multiply and infinitely connected Riemann surfaces, and even in the settings of one-parameter semigroups and of random dynamical systems (see, e.g., [H, L, B]). In several complex variables, the first Wolff-Denjoy theorems are due to Hervé [He1, 2]; in particular, in [He2] he proved a statement identical to the one above for fixed points free self-maps of the unit ball B n ⊂ C n . Hervé's theorem has also been generalized in various ways to open unit balls of complex Hilbert and Banach spaces (see, e.g., [BKS, S] and references therein).A breakthrough occurred in 1988, when the first author (see [A1]) showed how to prove a Wolff-Denjoy theorem for holomorphic self-maps of smoothly bounded strongly convex domains in C n . The techniques introduced there turned out to be quite effective in other contexts too (see, e.g., [A5, AR, Br1, Br2]); but in particular they led to Wolff-Denjoy theorems in smooth strongly pseudoconvex domains and smooth domains of finite type (see, e.g., [A4, Hu, RZ, Br3]).Two natural questions were left open by the previous results : how much does the boundary smoothness matter? And, what happens in weakly (pseudo)convex domains? As already shown by the results obtained by Hervé [He1] in the bidisk, if we drop both boundary smoothness and strong convexity the situation becomes much more complicated; but most of Hervé's techniques were specific for the bidisk, and so not necessarily applicable to more general domains. On the other hand, for smooth weakly convex domains a Wolff-Denjoy theorem was already obtained in [A3] (but here we shall get a better result; see Corollary 3.2).In 2012, Budzyńska [Bu2] (see also [BKR] and [Bu3] for infinite dimensional generalizations) finally proved a Wolff-Denjoy theorem for holomorphic fixed point free self-maps of a bounded strictly convex domain in C n , under no smoothness assumption on...

show abstract