20.500.12556/RUL-140686 Osnove polinomske kompleksne dinamike delo diplomskega seminarja Introduction to complex polynomial dynamics 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. 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. iterati kompleksnih polinomov napolnjena Juliajeva množica Böttcherjev izrek iterates of complex polynomials filled Julia set Böttcher's theorem true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2022-09-17 08:15:04 2022-09-17 08:15:07 2024-05-29 12:08:24 0000-00-00 00:00:00 2022 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517 127393 122321411 5974.pdf 5974.pdf 1 9D1FE82D04CFD109F530B216FE28B6ED 05937a981d0e3459f1d6b433b3213a0f7a929505486174a0dcc2a5783f29c74a 01e4f0e2-3650-11ed-92af-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=161479 Fakulteta za matematiko in fiziko 0 0 0