Qualitative Theory of Differential Equations: Although during the years, we have worked on so many different problems related to the qualitative theory of differential equations, during the last few years we have centered most of our interest in problems surrounding the well-known 16th Hilbert′s problem and the Jacobian Conjecture. We bring out our work on the Markus-Yamabe conjecture, Liapunov constants, center-focus problem, Hamiltonian perturbations, Abelian integrals, isochronicity, uniqueness of limit cycles, etc. Other kinds of problems we are also interested in are the general theory of polynomial differential systems, control theory, the study of singularities via blow-ups, nondifferentiable dynamical systems, structural stability, quadratic polynomial systems, etc. Some collaborators are C. Christopher, F. Dumortier, J.P. Françoise, H. Giacomini, C. Li, D. Shafer, J. Sotomayor, M.A. Teixeira, and X. Zhang.
Discrete Dynamical Systems: We are mainly interested in the study of the dynamics of continuous maps of one-dimensional topological spaces. Our main subjects of interest in this field are Periodic structure, Combinatorial Dynamics, Topological dynamics, and Topological entropy. In some special cases (including higher dimensional spaces), when studying the periodic structure, we also use the general theory of Lefschetz and Nielsen. We are also concerned about the study of the minimal dynamics in a homotopy class. Some colaborators are J. Guaschi, S. Kolyada, J. Los, M. Misiurewicz, J. A. Rodríguez, L. Snoha, and X. Ye.
Celestial Mechanics: The main goal of our research is the description of the global flow for certain restricted three-body problems by using symbolic dynamics. We emphasize the dynamics near homoclinic and heteroclinic orbits. We also contribute to the study of the number and distribution of the central configurations for the n-body problem. Indeed we use all this information to study the dynamics and the global flow associated with the n-body problem. Some collaborators are M. Falconi, E.A. Lacomba, and E. Perez-Chavela.
Complex Dynamics: Our general goal is to describe the dynamical and parameter plane given by the iteration of holomorphic maps of the complex plane or the Riemann sphere. In particular, we focus our attention on transcendental entire or meromorphic families (standard map, complex exponential family, Newton method applied to entire maps, etc) or rational iteration. In any of those settings, our approach is mostly topological including the description of the Julia set, the study of the boundary of Siegel discs, the distribution of Herman rings, the classification of the Baker domains, the properties of the bifurcation locus, etc. The main tools we use are quasiconformal surgery, symbolic dynamics, and complex analysis. Some collaborators are K. Baranski, X. Buff, B. Branner, R. Devaney, L. Geyer, and M. Moreno-Rocha.
