The Denjoy–Wolff theorem in

Budzyńska, Monika

doi:10.1016/j.na.2011.05.064

Cited by 23 publications

(15 citation statements)

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“…Busemann sequences have been introduced (under the name horosphere sequences) in balanced bounded convex domains in [28], and later studied in bounded convex domains in [14]. The proof of the following result is an adaptation of the ideas contained in those papers.…”

Section: Admissible Sequences Busemann Sequences and Horosphere Bounmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Horosphere topology

Bracci¹,

Gaussier²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains.

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Section: Admissible Sequences Busemann Sequences and Horosphere Bounmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Horosphere topology

Bracci¹,

Gaussier²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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“…In a Hilbert space ellipsoids replace the internally tangent discs [18]. In strictly convex domains, in C n [1,2,7], or in Banach spaces [9,10], the internally tangent discs are replaced by horospheres defined in terms of the Kobayashi distance. Although these horospheres are defined for arbitrary Banach spaces [2,3,32], if the boundary of the ball is more complicated they are considerably less tractable, even in finite dimensions [17,4,16].…”

Section: Introductionmentioning

confidence: 99%

“…Although these horospheres are defined for arbitrary Banach spaces [2,3,32], if the boundary of the ball is more complicated they are considerably less tractable, even in finite dimensions [17,4,16]. While holomorphic iteration on both finite and infinite dimensional Banach spaces has recently continued apace [5,7,8,9,11,12,20,28,31,32,34], spaces whose balls have, for example, non-strictly convex boundaries, even classical spaces such as the C * -algebras or L(H, K), still require a Wolff theorem and concrete descriptions of invariant domains to facilitate progress on iteration.…”

Section: Introductionmentioning

confidence: 99%

A Wolff Theorem for finite rank bounded symmetric domains

Mellon

2017

Journal of Mathematical Analysis and Applications

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We present a Wolff Theorem for all infinite dimensional bounded symmetric domains of finite rank. Namely, if B is the open unit ball of any finite rank JB *triple and f : B → B is a compact holomorphic map with no fixed point in B, we prove convex f-invariant subdomains of B (of all sizes and at all points) exist in the form of simple operator balls c λ + T λ (B), for c λ ∈ B and T λ an invertible linear map. These are exact infinite dimensional analogues of the invariant discs in ∆, the invariant ellipsoids in the Hilbert ball and invariant domains in finite dimensional triples. Results are new for rank > 2, even for classical spaces such as C *-algebras and JB *-algebras.

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“…Small horospheres might be too small; as shown by Frosini [F], there are holomorphic self-maps of the polydisk with no invariant small horospheres. We thus need another kind of horospheres, defined by Kapeluszny, Kuczumow and Reich [KKR2], and studied in detail by Budzyńska [Bu2]. To introduce them we begin with a definition:…”

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confidence: 99%

Wolff–Denjoy theorems in nonsmooth convex domains

Abate

Raissy

2013

Annali di Matematica

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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...

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The Denjoy–Wolff theorem in

Cited by 23 publications

References 30 publications

Horosphere topology

Horosphere topology

A Wolff Theorem for finite rank bounded symmetric domains

Wolff–Denjoy theorems in nonsmooth convex domains

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