Dinamika kompleksnih analitičkih funkcija

master's thesis

• Title (english): Dynamics of complex analytical functions
• Author: Dubravka Božić
• Mentor: Sonja Štimac (mentor)
• Committee member: Sonja Štimac (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2013-04-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom dijelu rada definiramo kompleksnu analitičku funkciju, navodimo osnovne rezultate kompleksne analize, topološke dinamike i upoznajemo se sa pojmom Riemannove sfere. Takoder, klasificiramo fiksne, odnosno periodičke točke čije orbite proučavamo u narednim poglavljima. Najvažniju ulogu u dinamici kompleksnih analitičkih funkcija imaju orbite odbojnih periodičkih točaka. Juliaov skup definiramo kao zatvorenje skupa svih odbojnih periodičkih točaka polinoma P, te dokazujemo da je on skup svih točaka u kojima familija iteracija polinoma P nije normalna familija. Iz dokaza tog teorem slijede gotovo sva svojstva Juliaovih skupova. Tako naprimjer vrijedi da je Juliaov skup polinoma P perfektan, potpuno invarijantan, neprazan skup, te da J(P) ima prazan interior. Vrijedi i da je J(P) zatvarač skupa svih praslika točke \(z_0\) iz J(P). Posebno, Juliaov skup je taj na kojemu polinom P ima kaotičnu dinamiku.

Abstract (english)

In the first part of this paper we defined the complex analytic function, we outline the basic results of complex analysis, topological dynamics, and we are introduced to the concept of Riemann sphere. Also, we classify fixed (periodic) points whose orbits are studied in the following chapters. The most important role in the dynamics of complex analytic functions is played by the orbits of repelling periodic points. We defined the Julia set as the closure of the set of repelling periodic points of a polynomial P and give a proof that this set is a set of points at which the family of iterates of P fails to be normal family. Almost all of the properties of the Julia sets follow from that theorem. Thus, for example, the Julia set of a polynomial P is a perfect, completely invariant, nonempty set which has an empty interior. It is walid that J(P) is the closure of all preimages for point \(z_0\) from J(P). In particular, the Julia set is the set which carries the chaotic dynamics of polynomial P.

• Keywords: Riemannova sfera; Juliaov skup; kaotična dinamika; Cantorov skup; zatvoreni interval; bazilika; dendrit; Mandelbrotov skup; Fatouov skup
• Keywords (english): Riemann sphere; Julia set; chaotic dynamics; Cantor set; closed interval; basilica; dendrite; Mandelbrot set; Fatou's set
• Language: croatian
• URN:NBN: urn:nbn:hr:217:060676
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-16 11:45:21