Dynamic in low dimensions

Abstract

The modern theory of dynamical systems started with the work of Poincaré and, since then, grew into a mature and very active branch of mathematical research. The main goal of this project is to further the study of the following areas of dynamical systems theory: - Hamiltonian systems with two degrees of freedom, their dynamical and topological aspects. - Polinomial differential equations on the plane and the 16th Problem of Hilbert. - Two¬dimensional homeomorphisms and diffeomorphisms such as Hénon maps and twits maps of the annulus. Renormalization theory in dimensions 1 and 2. - Interval endomorphisms (e.g., delicate analytic questions such as decay of geometry and existence of invariant measures); critical circle mappings; renormalization and parameter space. - Teichmueller theory and connections with low dimensional dynamics. - Differentiable ergodic theory. While dynamical systems theory developed it also moved away from other branchs of mathematics which were also started by Poincaré: symplectic geometry and topology. Another important goal of this project is to look for and to develop connections between these areas to the point of making it possible to use techniques of each area to attack problems of the other. (AU)