Wolff–Denjoy theorems in nonsmooth convex domains

Abstract

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...

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

“…(1) f has a fixed point in D; or In case D is a bounded strictly convex domain, a Denjoy-Wolff theorem of the previous type has been proved by Budzyńska [19], while, for bounded C 2 -smooth strictly C-linearly convex domains, the result is due to Abate and Raissy [6].…”

A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems

“…For various reasons having to do with their intrinsic geometry, convex domains predominate among recent generalizations of the Wolff-Denjoy theorem: see, for instance, [8,11,4,29] and several of the results in [20]. Visibility in the sense of Definition 1.1 is one of the key ingredients in the proof by Bharali-Zimmer of a generalization [9, Theorem 1.10] of Result 1.7 to taut Goldilocks domains.…”

Horosphere topology

“…In this paper we introduce a completely new prime ends theory defined via horospheres related to sequences. Horospheres have been used pretty much in geometric function theory in one and several variables, especially for studying iteration theory, Julia's Lemma, Denjoy-Wolff theorems (see, e.g., [1,3,4,14,15,37,21] and references therein), and they are a particular instance of a general notion of horospheres in locally complete metric spaces, see [17,6]. In complex geometry, horospheres defined by using complex geodesics are sometimes called Busemann horospheres.…”