Time: August , 21 9:30-11:30; August, 23 9:30-11:30
Abstract : In the conservative, smooth setting, a map has positive metric entropy iff it displays a positive Lyapunov exponent on a set of positive Lebesgue measure. A main question is that of their typicality among surface dynamics.
I will begin this lecture by a state of the art together with a list of related, open questions and conjectures.
Then I will focus on a theorem we proved recently with Dimtry Turaev, the Herman's positive entropy conjecture:
Any smooth, sympletic diffeomorphism $f\in Diff^\infty_\omega(M^2)$ of a sympletic surface $(M^2, \omega)$ which displays a non-hyperbolic periodic point can be $C^\infty_\omega$-approximated by one with positive metric entropy.
In the last part of this mini-course I will explain in more details the technics involved:
- Surgery for Anosov after Katok, Przytycki, Arroyo-Pujals, B.-Turaev.
- normal form nearby saddle point and renormalization.
- Universal Dynamics after Turaev, and development done in our work.
- Restoration of heteroclinic links
