Two problems in the local geometry of real analytic dynamical systems. Planar diffeomorphisms and three dimensional hopf vector fields

• Autores: María Martín Vega
• Directores de la Tesis: Nuria Corral Pérez (dir. tes.) , Fernando Sanz Sánchez (codir. tes.)
• Lectura: En la Universidad de Valladolid ( España ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: Rafael Ortega Ríos (presid.) , Helena Reis (secret.) , David Marín Pérez (voc.)
• Enlaces
  • Tesis en acceso abierto en: UVADOC
• Resumen

  • This PhD thesis can be framed in the Local Geometry of Real Analytic Dynamical Systems. We deal with two different (but related) problems concerning this topic. The first problem deals with the discrete dynamical systems generated by the iteration of the germ of a two dimensional real analytic diffeomorphism that fixes a point. The second problem deals with the dynamics generated by the flow of the germ of three dimensional real analytic singular vector field.

    Roughly, the two principal results that we present in this text are the following ones: Problem I. Sectorial decomposition of germs of tangent to the identity real analytic plane diffeomorphisms. Under the (necessary) hypothesis that F is of "non center-focus type", we prove that there exists a partition of a neighborhood U of the fixed point into a finite number of topological submanifolds, such that the orbits of F on each submanifold have a uniform well-established asymptotic behavior. As a consequence, we obtain that the set of periodic points of F in U coincides with the set of fixed points. Under some non-degeneracy conditions (that hold in particular when the fixed point is an isolated fixed point of F), U can be assumed to be a semi-analytic open set and that each stratum is a real analytic submanifold.

    Problem II. Description of the local cycle locus and Dulac's problem for germs of real analytic vector fields at the origin of R^3. Here we take such germs with a Hopf singularity at 0, i.e. whose linear part has two conjugated purely imaginary eigenvalues. For these vector fields, we prove that the union of all the cycles (periodic trajectories) in a sufficiently small neighborhood of 0 is empty, or equal to a finite number of subanalytic surfaces, or a dense open set (in fact the compliment of the singular locus). We also give a characterization of the last situation in terms of analytic linearization of the foliation generated by the vector field and in terms of complete integrability. As a consequence, we obtain that there cannot exist infinitely many isolated cycles accumulating and collapsing to 0 (Dulac's property).