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.
