Linearizacija holomorfnih funkcij v bližini negibnih točk

Gombač, Matevž (avtor), Boc Thaler, Luka (mentor)

Kompleksna dinamika se ukvarja z iteriranjem holomorfnih funkcij na Riemannovi sferi. V diplomskem delu so natančneje predstavljeni dinamični sistemi, ki jih dobimo z iteriranjem racionalnih funkcij oziroma iteriranjem holomorfnih endomorfizmov Riemannove sfere f : CP^1-> CP^1. Družina iteratov {f^n; n naravno število} razdeli CP^1 na komplementarni si množici; Fatoujevo množico F, tj. množico točk, v okolici katerih je družina {f^n; n naravno število} normalna, ter Juliajevo množico J, ki je njen komplement. Vsaka taka preslikava ima negibne točke, ki jih delimo na pet tipov glede na njihov množitelj. Glavni cilj diplomskega dela je lokalno linearizirati f v okolici negibnih točk in opisati obnašanje orbit v njihovi bližini. Dokažemo, da je lokalno v bližini negibne točke f konformno konjugirana linearni preslikavi z |-> kz v primeru privlačnih, odbojnih in nekaterih iracionalno nevtralnih negibnih točk. V primeru superprivlačnih negibnih točk je f konjugirana z |-> z^p, pri racionalnih pa lahko na nekem disku, ki ima negibno točko na svojem robu, funkcijo f konjugiramo do translacije z|-> z+1.

• Jezik: Slovenski jezik
• Ključne besede: Negibna točka, konformna konjugacija, množitelj, normalna družina, Fatoujeva in Juliajeva množica
• Vrsta gradiva: Diplomsko delo/naloga
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2024
• Datum objave v RUL: 11.09.2024

Sekundarni jezik

• Jezik: Angleški jezik
• Naslov: Linearization of holomorphic functions near fixed points

Izvleček

Complex dynamics is the study of iteration of holomorphic functions on the Riemann sphere. In this thesis, we will focus on dynamical systems obtained by iterating rational functions, that is iterating a holomorphic endomorphism of the Riemann sphere f : CP^1 -> CP^1. The family of iterates {f^n; n natural number} divides CP^1 into two complementary sets; the Fatou set F , that is the set of points in the neighbourhood of which the family {f^n; n natural number} is normal, and the Julia set J, which is its complement. Each such map has fixed points, which are classified into five types according to their multiplier. The main goal of this thesis is to locally linearize f in some neighbourhood of fixed points and describe the behaviour of orbits in their vicinity. We prove that locally near a fixed point, f is conformally conjugate to a linear map z |-> kz in the case of attracting, repelling and some irrationally indifferent fixed points. In the case of superattracting fixed points f is conjugate to z |-> z^p, and in the case of rationally indifferent ones it is conjugate to z |-> z+1 in some disk with the fixed point on its boundary.

• Ključne besede: Fixed point, conformal conjugation, multiplier, normal family, Fatou and Julia set