Homeomorphic extension of Quasi-Isometries and Iteration Theory

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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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
Corso di studio
Curriculum A: Generale e applicativo
Ordinamento Cds
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

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