Avances en el estudio de propiedades topológicas de flujos analíticos sobre superficies

Resum

In broad terms, this dissertation intends to investigate the effect of the analyticity on the the field of continuous dynamical systems of dimension 2, that is, what dynamical phenomena appear when the function defining such a system is analytic. Being more concrete, we aim to advance in the study of the topological nature of the flows associated to real analytic vector fields on surfaces with the following four objectives. OBJECTIVE 1. The topological classification of unstable global attractors for flows associated to polynomial vector fields on the plane. OBJECTIVE 2. The topological characterization of the omega-limit sets for flows associated to analytic vector fields on open subsets of the sphere and the projective plane. OBJECTIVE 3. The topological characterization of the limit periodic sets for families of flows associated to polynomial vector fields on the plane. OBJECTIVE 4. The study of the flows associated to analytic vector fields on surfaces with the property of having all their orbits dense (the so-called minimal flows). Regarding the methodology, the research was carried out in the Department of Mathematics of the University of Murcia, under the support of the Research Group of Dynamic Systems of the Region of Murcia. The student has actively participated in the scientific activities developed by this group and has presented his work to the scientific community in different congresses (via oral presentations and posters). At the same time, the student has enjoyed short research visits at the University of Toronto in Canada, the University of Toulouse in France and the Instituto de Ciencias Matemáticas in Madrid (ICMAT) and two long stays (four months each) at ICMAT itself and at the University of Plymouth in the United Kingdom. This last stay allows to apply for the Mención Internacional de Doctorado. The research has led to the presented PhD Thesis, which attacks the previous four objectives, compiling research papers that the author of the Thesis has done in collaboration with its director, Victor Jiménez, as well as André Belotto, Daniel Peralta-Salas and Gabriel Soler. In collaboration with Víctor Jiménez, it has been possible to give the complete topological classification referred to in the first of the objectives above; in the same way, in collaboration with André Belotto, the topological characterization that corresponds to the second of the objectives has been completed. In a joint work with Daniel Peralta-Salas and Gabirel Soler, the fourth of the objectives was attacked, resulting in a complete classification of all orientable and non-orientable surfaces of finite genus that admit minimal analytical flows and presenting an example of non-orientable surface of infinite genus with the same property. Finally, in relation to the third objective, and also in collaboration with Víctor Jiménez, it has been possible to complete the proof of the topological characterization of the omega-limit sets for analytical flows on the sphere, the plane and the projective plane as well as to advance in the characterization of these sets for analytical flows defined on open subsets of the plane.