IFS fraktali

master's thesis

• Author: Martina Marušić
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-03-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Vizualno kompleksni oblici s infinitezimalno finom strukturom već su neko vrijeme predmet istraživanja matematičara. Ovaj rad bavi se posebnim tipom fraktala - IFS fraktalima koji se mogu koristiti za reprezentaciju raznih detaljnih samosličnih oblika u prirodi, kao što su oblaci, drveće, planine, obale mora, itd. Krećemo s definicijom fraktala u najopćenitijem smislu. Zatim prelazimo na iterirane funkcijske sustave (IFS). Definiramo ih, dajemo definiciju IFS fraktala, najvažnije primjere te algoritam za generiranje IFS fraktala. Također navodimo Kolaž teorem koji nam govori da ako želimo pronaći IFS čiji atraktor "nalikuje" danom skupu, moramo nastojati pronaći skup transformacija čija je unija, odnosno kolaž, slika danog skupa što bliže tom danom skupu. U sljedećem poglavlju uvodimo dinamičke sustave, njihovu definiciju i neke primjere, te ih povezujemo s iteriranim funckijskim sustavima. Dajemo algoritam za generiranje orbite dinamičkih sustava, te Shadowing teorem koji je značajan u praksi za računanje orbita dinamičkih sustava. Na kraju obrađujemo iterirane funkcijske sustave s vjerojatnostima (IFSP). Definiramo ih, dajemo primjere fraktala generiranih IFS-om s vjerojatnostima te algoritam za njihovo generiranje. Predstavljamo Eltonov teorem koji formalizira našu intuiciju o mjeri na fraktalima. Završavamo s teoremom koji rasvjetljava vezu između IFS-a i IFSP-a.

Abstract (english)

Visually complex shapes with infinitesimally fine structure have for some time now been a topic of research for mathematicians. This thesis focuses on a particular type of fractals - IFS fractals which can be used to represent a variety of detailed self-similar shapes in nature, such as clouds, trees, mountains, sea shores, etc. We begin with a definition of fractals in the most general sense. We then move to iterated function systems (IFS). We define them, give a definition of IFS fractals, the most important examples and an algorithm for generating IFS fractals. Furthermore, we state Collage theorem which says that in order to find an IFS whose attractor ’resembles’ a given set, we need to aim to find a set of transformations whose union, that is, collage, of images of the given set is as close as possible to the given set. In the next section we introduce dynamical systems, their definition and some examples, and we connect them to iterated function systems. We give an algorithm for generating an orbit of dynamical systems, and also state the Shadowing theorem which is important in practice for calculating orbits of dynamical systems. Lastly, we cover iterated function systems with probabilities (IFSP). We define them, give examples of fractals generated with an IFS with probabilities as well as an algorithm for generating them. We state Elton’s theorem which formalizes our intuition about measure on fractals. We end the thesis with a theorem which illuminates the connection between IFS and IFSP.

• Keywords: fraktali; IFS fraktali; iterirani funkcijski sustavi; dinamički sustavi
• Keywords (english): fractals; IFS fractals; iterated function systems; dynamical systems
• Language: croatian
• URN:NBN: urn:nbn:hr:217:331237
• 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; Embargo expiration date: 2021-08-29
• Created on: 2019-08-29 12:17:18