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

- Series
- School of Mathematics Colloquium
- Time
- Thursday, October 30, 2014 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Professor Albert Fathi – ENS-Lyon &amp; IUF

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.