Abstract

At the beginning of the 20th century, the celebrated Poincare-Birkhoff Theorem states the existence of periodic points for the twist map on Annulus. It has motivated a large amount of modern mathematics. Symplectic geometry, contact geometry, Seiberg-Witten theory, Arnold conjecture are examples. In this talk, we introduce some fine structure of periodic points for area-preserving diffeomorphisms on Annulus, two important dynamical invariants: the action and rotation number are introduced, we will discuss connections between the distribution of periodic points and these invariants in light of some new methods in contact geometry.