Lyapunov Functions: Towards an Aubry-Mather theory for homeomorphisms?

School of Mathematics Colloquium
Thursday, October 30, 2014 - 11:00am for 1 hour (actually 50 minutes)
Skiles 005
Professor Albert Fathi – ENS-Lyon & IUF
Martin Short
This is a joint work with Pierre Pageault. For a homeomorphism h of a compact space, a Lyapunov function is a real valued function that is non-increasing along orbits for h. By looking at simple dynamical systems(=homeomorphisms) on the circle, we will see that there are systems which are topologically conjugate and have Lyapunov functions with various regularity. This will lead us to define barriers analogous to the well known Peierls barrier or to the Maסי potential in Lagrangian systems. That will produce by analogy to Mather's theory of Lagrangian Systems an Aubry set which is the generalized recurrence set introduced in the 60's by Joe Auslander (via transfinite induction) and a Maסי set which is essentially Conley's chain recurrent set. No serious knowledge of Dynamical Systems is necessary to follow the lecture.