Research


Main Research Lines

  • Low Dimensional Flows and Hamiltonian Systems:

    • Dynamical and topological aspects of flows obtained by restricting Hamiltonian systems with two degrees of freedom to constant energy levels. Hofer conjecture. Geodesic flows on the two dimensional sphere.

    • Local properties of equilibrium points of simple mechanical systems (kinetic energy + potential energy).

  • Low Dimensional Discrete Dynamics:

    • Homeomorphisms and diffeomorphisms of the annulus, the cylinder and the torus. Rotation sets and periofics orbits. Boyland's conjecture.

    • Twist mappings. Rupture of invariant curves. Ergodicity of twist mappings on the torus.

    • Families of diffeomorphisms on surfaces. Hénon mappings. Implications among orbits and quasi-linear models in two dimensions.

  • Renormalization in one and two dimensions:

    • Renormalization of critical circle mappings and dissipaive mappings with one discontinuity. Complete families and circle endomorphisms. Geometry decay in cubic families. Henon renormalization.

    • Complex dynamics. Thompson groups and Teichmüller spaces. Dynamics of transcendent holomorphic and meromorphic functions.

  • Differentiable Ergodic Theory:

    • Small scale structure and non-stationary dynamics.

    • Infinite measures and fractal geometry.

    • Renormalization and Teichmüller flow.

  • 3-dimensional geometry and topology and connections with surface dynamics

  • Holomorphic dynamics in one and several dimensions.




Henon map