On Holomorphic Maps Which Commute with Hyperbolic Automorphisms

Fabritiis, Chiara de; Gentili, Graziano

doi:10.1006/aima.1998.1806

Cited by 7 publications

(7 citation statements)

References 8 publications

(4 reference statements)

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“…To generalize the construction of annuli to several complex variables, we will focus our attention to the action of hyperbolic elements on B n . First of all, we recall a result which is due to de Fabritiis and Gentili (see [5]).…”

Section: Preliminaries and Statementsmentioning

confidence: 96%

Complex geometry of generalized annuli

2003

Advg

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We study the complex geometry of a class of domains in C n which generalize the annuli in C, i.e., which are quotients of the unit ball B n of C n for the action of a group generated by a hyperbolic element of Aut B n. In particular, we prove that the degree of holomorphic maps between two such domains is bounded by a constant which depends on the ''radii'' of the domains only and we give some results on the existence of complex geodesics for the Kobayashi distance in these domains.

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Section: Preliminaries and Statementsmentioning

confidence: 96%

Complex geometry of generalized annuli

2003

Advg

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“…A word-by-word translation of the previous proof allows the following theorem (first discovered by de Fabritiis and Gentili [11]): Theorem 4.2 Let B n be the unit ball of C n . Let γ be an hyperbolic automorphism of B n and let f be a holomorphic self-map of B n which commutes to γ.…”

Section: Using Property J In Old Contextsmentioning

confidence: 99%

“…Our proof of Heins' Theorem is not simpler than the original one (see [14]), but it has the value that it doesn't require any explicit knowledge of the automorphisms group of ∆. For this reason our proof can be translated to prove an analogous result in the unit ball B n (see also [11] and [10]).…”

Section: Using Property J In Old Contextsmentioning

confidence: 99%

Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity

Bracci¹

2000

Can. math. bull.

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“…Uniqueness (up to composition with linear fractional maps) of intertwining mappings in higher dimension-without assign further conditions-does not hold. The main theoretical reason is that in dimension one the centralizer of a given hyperbolic automorphism consists of hyperbolic automorphisms while in higher dimension this is not longer so (see [8]). For example, if H : B N → B N is a hyperbolic automorphism, then any holomorphic self-map…”

Section: Further Remarks and Open Questionsmentioning

confidence: 99%

Valiron's construction in higher dimension

Bracci,

Gentili,

Poggi-Corradini

2007

Preprint

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We consider holomorphic self-maps ϕ of the unit ball B N in C N (N = 1, 2, 3, ...).In the one-dimensional case, when ϕ has no fixed points in D := B 1 and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map φ, and therefore, in this case, the dynamical properties of φ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on ϕ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation σ, which maps the ball into the right half plane of C, and solves the functional equation σ • ϕ = λσ, where λ > 1 is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of ϕ.

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On Holomorphic Maps Which Commute with Hyperbolic Automorphisms

Cited by 7 publications

References 8 publications

Complex geometry of generalized annuli

Complex geometry of generalized annuli

Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity

Valiron's construction in higher dimension

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