Abstract

The concept of topological entropy plays a central role in the modern theory of Dynamical Systems. Originally introduced by Adler, Konheim and McAndrew in 1960’s, it was later reformulated and clarified by work of Rufus Bowen. It has roots in the work of Kolmogorov and Sinai. Positive topological entropy is usually taken as the mathematical definition of (topological) chaos, in fact fundamental results of Katok imply that, in low dimensions, positive topological entropy forces the presence of horseshoes.

After discussing this circle of ideas, the goal of this talk will be to understand how variational methods can be used to detect positive topological entropy in Hamiltonian systems. The prototypical example is that of a geodesic flow on a negatively curved surface. In a second step we will discuss how sophisticated variational methods - which go under the umbrella of Symplectic Field Theory and are based on elliptic PDEs - can be used to generalise the same kind of results to larger classes of Hamiltonian systems.