Kvazikonformna kirurgija in Hermanovi kolobarji : delo diplomskega seminarja

Učakar, Beno (author), Kuzman, Uroš (mentor)

Kompleksna dinamika je področje matematike, ki se ukvarja z dinamičnimi sistemi določenimi z iteracijo kompleksnih preslikav. Cilj diplomske naloge je konstruirati racionalno preslikavo, katere dinamika je na 2-povezanem območju biholomorfno ekvivalentna iracionalni rotaciji. Tak fenomen imenujemo Hermanov kolobar. Najprej se seznanimo z osnovami kompleksne dinamike in opredelimo pojem Hermanovega kolobarja. Nato razvijemo teorijo kvazikonformnih preslikav in s pomočjo kvazikonformne kirurgije kostruiramo racionalno preslikavo, ki premore Hermanov kolobar.

• Language: Slovenian
• Keywords: kompleksna dinamika, negibna točka, Juliajeva množica, Fatoujeva množica, Sieglov disk, Hermanov kolobar, kvazikonformna preslikava, Beltramijeva enačba, eliptično polje, kvazikonformna kirurgija
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-138742
• UDC: 517
• COBISS.SI-ID: 118585859
• Publication date in RUL: 12.08.2022

Secondary language

• Language: English
• Title: Quasiconformal surgery and Herman rings

Abstract

Complex dynamics is the study of dynamical systems defined by the iteration of complex valued functions. The goal of this thesis is to construct a rational function with dynamics on a 2-connected region biholomorphically equivalent to an irrational rotation. This phenomenon is called a Herman ring. First we familiarize ourselves with the basics of complex dynamics and define the term Herman ring. Then we develop the theory of quasiconformal maps and with the help of quasiconformal surgery construct a rational function with a Herman ring.

• Keywords: complex dynamics, fixed point, Julia set, Fatou set, Siegel disk, Herman ring, quasiconformal mapping, Beltrami equation, elliptic field, quasiconformal surgery