Fatou componentsBoc Thaler, Luka (Avtor) Forstnerič, Franc (Mentor) Fornæss, John Erik (Komentor) holomorphic functionsStein manifoldslong ▫$\mathbb{C}^n$▫complex dynamicspolynomialsentropyseveral complex variablesFatou-Bieberbach domainlimit setsIn this thesis we address some problems in complex dynamics and classical complex analysis of several variables. Chapter I provides a historical background of the field of complex dynamics. Main results regarding the dynamics of complex rational functions are discussed and some motivation for generalizing this theory to higher dimensions is given. In Chapter II we study the regularity of Fatou components for holomorphic endomorphisms of ▫$\mathbb{P}^k$▫. We show that for ▫$k=1$▫ all Fatou components are regular and that this is not true in general. Next we study the difference between the Julia set and the support of the equilibrium measure. We prove that either they coincide or else the support of the equilibrium measure is nowhere dense in the Julia set. It follows from our results that if this two sets coincide, then all Fatou components are regular. We give an example of a bounded Stein domain in ▫$\mathbb{P}^2$▫ whose regularization is not Stein. Chapter III is a joint work with Han Peters and John Erik Fornæss. We study invariant Fatou components for holomorphic endomorphisms in ▫$\mathbb{P}^2$▫. In the recurrent case these components were classified by Fornæss and Sibonny. In 2008 Ueda completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of ▫$\mathbb{C}^2$▫ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets. In Chapter IV we focus on the complex manifolds which can be exhausted by copies of ▫$\mathbb{C}^n$▫, and are therefore called long ▫$\mathbb{C}^n$▫'s. This manifolds have a connection to an old "union problem" which was solved in 1976 by J.E. Fornæss. In 2010 Wold has constructed a non-Stein long ▫$\mathbb{C}^n$▫ but the question was left open whether there are more then just one non-Stein long ▫$\mathbb{C}^n$▫ and does there exist one without any non-constant holomorphic functions. We answer to this questions affirmatively and we provide some more results. It is still a wide open problem whether or not ▫$\mathbb{C}^n$▫ is the only Stein long ▫$\mathbb{C}^n$▫. The main results were obtained in the conversation with Franc Forstnerič. Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. In Chapter V we show that their results also hold for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the ▫$2m + 1$▫-st image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.[L. Boc Thaler]20162017-09-22 02:54:11Doktorsko delo/naloga95864sl