Osnove polinomske kompleksne dinamike : delo diplomskega seminarja

Tiselj, Peter (author), Prezelj, Jasna (mentor)

Preslikave kompleksnih števil $\mathbb{C} \rightarrow \mathbb{C}$ pomenijo slikanje iz ene dvodimenzionalne ravnine v drugo. Zato graf vsakršne funkcije iz $\mathbb{C} \rightarrow \mathbb{C}$ leži v štiridimenzionalnem prostoru, kar pomeni, da imamo v našem tridimenzionalnem svetu težave z vizualizacijo takšnih funkcij. V diplomski nalogi obravnavamo in poizkušamo predstaviti kompleksne preslikave tipa $q_c(z) =z^2+c$, $c \in \mathbb{C}$ in $R(z) = z^n+P(z)/Q(z)$, $n \geq 2$, stopnja polinoma $Q$ pa večja ali enaka stopnji polinoma $P$. Natančneje, zanimala nas bo konvergenca funkcijskega zaporedja iteratov dane funkcije; iterirati $f$ pomeni s funkcijo $f$ zaporedno delovati na isti vhodni podatek. Napolnjeno Juliajevo množico, označeno s $K(f)$, tvorijo tista kompleksna števila, ki pod iteracijo $f$ ostanejo omejena. Da imamo opraviti s tridimenzionalnimi objekti, definiramo množici $U(f)$ oz. $V(f)$, v katerih so tista kompleksna števila, katerih realni oz. imaginarni deli iteratov pod funkcijo $f$ so omejeni. Ugotovimo, da je za $q_c(z)$ napolnjena Juliajeva množica $K(q_c)$ enaka $U(q_c)$ in da velja $K(q_c) \neq V(q_c)$. Za družino $R(z)$ pokažemo, da je $K(R) = U(R)$, če je $n$ sod, $K(R) \neq U(R)$, če je $n$ lih in $K(R) \neq V(R)$. Kot primer uporabe smo si v zadnjem delu ogledali iskanje ničel v kompleksnem z Newtonovo metodo in opazovali območja privlaka.

• Language: Slovenian
• Keywords: iterati kompleksnih polinomov, napolnjena Juliajeva množica, Böttcherjev izrek
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-140686
• UDC: 517
• COBISS.SI-ID: 122321411
• Publication date in RUL: 17.09.2022

TISELJ, Peter, 2022, Osnove polinomske kompleksne dinamike : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 26 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=140686

Secondary language

• Language: English
• Title: Introduction to complex polynomial dynamics

Abstract

A complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ can be viewed as mapping from two dimensional real plane to itself. Consequently, a graph of such function lies in a real four dimensional space, which means that it is hard to visualize in our three dimensional world. In my thesis we investigate the behaviour and visualization of complex maps $q_c(z) =z^2+c$, $c \in \mathbb{C}$ and $R(z) = z^n+P(z)/Q(z)$, $n \geq 2$, where the degree of polynomial $Q$ is greater or equal to the degree of the polynomial $P$. More precisely, we examine the convergence of the functional sequence of iterates of a given function; to iterate $f$ is to apply $f$ repeatedly to an input. Those complex numbers which are bounded under iteration with $f$ form the filled Julia set, denoted as $K(f)$. We define sets $U(f)$ and $V(f)$ which contain the complex numbers with the real and imaginary components that remain bounded under iterations with the map $f$, respectively. We prove that for the $q_c(z)$ the filled Julia set $K(q_c)$ equals $U(q_c)$ and that inequality $K(q_c) \neq V(q_c)$ holds. For the family of maps $R(z)$ we show that $K(R) = U(R)$ if $n$ is even, $K(R) \neq U(R)$ if $n$ is odd; the inequality $K(R) \neq V(R)$ always holds. In the last section we examine more in detail the Newton method applied on holomorphic functions.

• Keywords: iterates of complex polynomials, filled Julia set, Böttcher's theorem