Study of the dynamics of some planar maps

Zusammenfassung

The aim of this doctoral thesis is to study the dynamics associated to certain planar maps, including the study of objects such as periodic orbits, continuous periodic points, rotation numbers and Lie symmetries, among others. More specifically, it is proved that any planar birational integrable map, which preserves a fibration given by genus 0 curves has a Lie symmetry and some associated invariant measures. The obtained results allow the study in a systematic way of the global dynamics of these maps. In particular, for concrete examples, it is given the explicit expression of the rotation number function, and the set of periods of the considered maps. There are also presented three methodologies to find a continua of periodic points with a prescribed period for rational maps having rational first integrals. The first two ones are known. In this work these methods are applied when the maps are birational and the generic level sets of the corresponding first integrals have either genus 0 or 1. The third methodology is new. Unlike the other methods, where the algebraic structure of the invariant curves is used, the new method is strictly computational, and it is based on the study of the resultant associated to the equations that characterize the periodic orbits and the invariant algebraic curves. On the other hand, some special Landen transformations are also studied. Several examples of Landen transformations are presented, and they are applied to the computation of some defined integrals. One of the considered examples is the rational integral formed by polynomials with even terms, with numerator of degree 4 and denominator of degree 6. This transformation has an associated dynamical system defined by a planar map studied by Boros and Moll (2000), and by Chamberland and Moll (2006). The different invariant regions of this dynamical system, and the relationship between the dynamics of the map and the convergence of the considered integral, are presented. For this dynamical system it is proved the existence of fixed points and 3-periodic points to see that a conjecture proposed in V.H. Moll in Numbers and functions. From a classical experimental mathematician’s point of view. Student Mathematical Library, 65 (American Mathematical Society, Providence RI 2012), is false. To prove it, it is developed a new methodology that consists in: to convert the characterization of the periodic points in an algebraic problem; the combination of an algorithm based on the Sturm's method to isolate all the real roots of one variable polynomial and a procedure for discarding possible solutions for polynomial equation systems; and the application of the Poincaré-Miranda theorem. Finally, it is done a numerical analytic study that provides evidences of the existence of homoclinic behaviors associated to one of the fixed points of the considered dynamical system, and also some points at the intersection of unstable variety of this fixed point and in the non-definite set of the dynamical system.