Igors Gorbovickis

This thesis focuses on several independent problems from analytic theory of dynamical systems and discrete geometry. The problems in analytic dynamical systems are split into two parts, first part related to local properties, and the second one related to global properties. In the beginning of the first part we prove a Poincaré-Dulac type theorem on resonant normal forms for families of maps analytically depending on a parameter. In the next Section we address the question: how many periodic orbits and of which periods may be born from a fixed point of a germ of a holomorphic map under a small perturbation? We give a complete answer to this question for a large class of linearization matrices of the map at the fixed point in the case when the dimension of the phase space is n ≤ 3. We show that the answer to this question is essentially different for n > 3. In the second part we prove that a typical volume preserving polynomial automorphism of C² does not have homoclinic tangencies between stable and unstable manifolds of any periodic point. We also show that if a certain condition holds, then a typical volume preserving polynomial automorphism of C² has a (more general) Kupka-Smale property. We give connection between this problem and the following question: consider the multiplier of a periodic orbit of a polynomial of one variable of degree d as an algebraic function of that polynomial obtained by analytic continuation of the periodic orbit. Are any two such functions algebraically independent over C? We give a positive answer to this question for d ≥ 3. In the last part of this thesis we study the Kneser-Poulsen conjecture from discrete geometry. The conjecture says that if a finite set of balls in Rⁿ is rearranged so that the distances between each pair of centers do not decrease, then the volume of the union (intersection) of the balls does not decrease (does not increase). We prove two theorems that confirm the conjecture under some additional assumptions. In the first theorem we assume that radii of the balls are all equal and sufficiently large compared to the maximal distance between a pair of centers. In the second theorem we assume that in the initial configuration each ball intersects no more than n + 2 other balls.