Bubani, Elia
(2021)
Homeomorphic extension of Quasi-Isometries and Iteration Theory.
[Laurea magistrale], Università di Bologna, Corso di Studio in
Matematica [LM-DM270]
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Abstract
Starting from the Riemann Mapping Theorem it arises the interest for
biholomorphisms over domains in one or several complex variables. Poincar´e
showed that there is no analytic isomorphism between the Polydisc and the
unit ball already in C^2. The previous fact may suggest that biholomorphic
domains are a class of such well-behaved sets that could extend some regularity of the biholomorpshism until their respective boundaries. A very influent
approach was faced by Fefferman (published in the year 1974), by
proving that every biholomorphism between bounded strongly pseudoconvex
domains with
smooth boundaries extends as a diffeomorphism to the closures of the domains. In this work is quoted a classical result that presents
an isometry respect to the Bergman metric between biholomorphic domains
and he noticed an interesting behaviour of geodesics when they are going to
the boundary of a considered domain.
The first part of this thesis mainly follows Abate’s work aiming to show the homeomorphic extension of a biholomorphism between C^2-smooth strongly pseudoconvex domains.
The second part of this thesis mainly follows the work of Bracci, Gaussier and
Zimmer aiming to show the homeomorphic extension of a Quasi-Isometric homeomorphisms to the End compactifications of the respective domains.
Other consequences are related to extend the Denjoy-Wolff
Theorem for domains in several complex variables and present the Denjoy-Wolff behaviour
for commuting holomorphic selfmaps with no fixed point in the domain itself.
Abstract
Starting from the Riemann Mapping Theorem it arises the interest for
biholomorphisms over domains in one or several complex variables. Poincar´e
showed that there is no analytic isomorphism between the Polydisc and the
unit ball already in C^2. The previous fact may suggest that biholomorphic
domains are a class of such well-behaved sets that could extend some regularity of the biholomorpshism until their respective boundaries. A very influent
approach was faced by Fefferman (published in the year 1974), by
proving that every biholomorphism between bounded strongly pseudoconvex
domains with
smooth boundaries extends as a diffeomorphism to the closures of the domains. In this work is quoted a classical result that presents
an isometry respect to the Bergman metric between biholomorphic domains
and he noticed an interesting behaviour of geodesics when they are going to
the boundary of a considered domain.
The first part of this thesis mainly follows Abate’s work aiming to show the homeomorphic extension of a biholomorphism between C^2-smooth strongly pseudoconvex domains.
The second part of this thesis mainly follows the work of Bracci, Gaussier and
Zimmer aiming to show the homeomorphic extension of a Quasi-Isometric homeomorphisms to the End compactifications of the respective domains.
Other consequences are related to extend the Denjoy-Wolff
Theorem for domains in several complex variables and present the Denjoy-Wolff behaviour
for commuting holomorphic selfmaps with no fixed point in the domain itself.
Tipologia del documento
Tesi di laurea
(Laurea magistrale)
Autore della tesi
Bubani, Elia
Relatore della tesi
Correlatore della tesi
Scuola
Corso di studio
Indirizzo
Curriculum A: Generale e applicativo
Ordinamento Cds
DM270
Parole chiave
homeomorphic extension Gromov Compactification Hyperbolicity Strongly Pseudoconvex domains Carathéodory distance Kobayashi distance Quasi-Isometry Shadowing Lemma Commuting 1-Lipschitz self-maps Denjoy-Wolff Theorem
Data di discussione della Tesi
26 Marzo 2021
URI
Altri metadati
Tipologia del documento
Tesi di laurea
(NON SPECIFICATO)
Autore della tesi
Bubani, Elia
Relatore della tesi
Correlatore della tesi
Scuola
Corso di studio
Indirizzo
Curriculum A: Generale e applicativo
Ordinamento Cds
DM270
Parole chiave
homeomorphic extension Gromov Compactification Hyperbolicity Strongly Pseudoconvex domains Carathéodory distance Kobayashi distance Quasi-Isometry Shadowing Lemma Commuting 1-Lipschitz self-maps Denjoy-Wolff Theorem
Data di discussione della Tesi
26 Marzo 2021
URI
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