20.500.12556/RUL-161402 Linearizacija holomorfnih funkcij v bližini negibnih točk delo diplomskega seminarja Linearization of holomorphic functions near fixed points 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 $f : {\mathbb C}_\infty \to {\mathbb C}_\infty$. Družina iteratov $\{f^n; n \ge 1\}$ razdeli ${\mathbb C}_\infty$ na komplementarni si množici; Fatoujevo množico $F$, tj. množico točk, v okolici katerih je družina $\{f^n; n \ge 1\}$ 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 $\zeta$ $f$ konformno konjugirana linearni preslikavi $z \mapsto \lambda z$ v primeru privlačnih, odbojnih in nekaterih iracionalno nevtralnih negibnih točk. V primeru superprivlačnih negibnih točk je $f$ konjugirana $z \mapsto z^p$, pri racionalnih pa lahko na nekem disku, ki ima negibno točko na svojem robu, funkcijo $f$ konjugiramo do translacije $z \mapsto z+1$. 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 $f : {\mathbb C}_\infty \to {\mathbb C}_\infty$. The family of iterates $\{f^n; n \ge 1\}$ divides ${\mathbb C}_\infty$ into two complementary sets; the Fatou set $F$ , that is the set of points in the neighbourhood of which the family $\{f^n; n \ge 1\}$ 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 $\zeta$, $f$ is conformally conjugate to a linear map $z \mapsto \lambda z$ in the case of attracting, repelling and some irrationally indifferent fixed points. In the case of superattracting fixed points f is conjugate to $z \mapsto z^p$, and in the case of rationally indifferent ones it is conjugate to $z \mapsto z+1$ in some disk with $\zeta$ on its boundary. Negibna točka konformna konjugacija množitelj normalna družina Fatoujeva in Juliajeva množica Fixed point conformal conjugation multiplier normal family Fatou and Julia set true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2024-09-11 08:15:14 2024-09-11 08:15:18 2024-09-17 08:37:54 0000-00-00 00:00:00 2024 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.5 142910 207757059 12825.pdf 12825.pdf 1 D9C3CDEE88D21F0E5076445D19098863 6e1b378ec02596cce3c334dda562d8e15d09ea05eb6bcbf2f5b81c70a7fcaa4a 2d7fa866-7005-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=190331 Fakulteta za matematiko in fiziko 0 0 0