Workshop on Recent Developments in Symplectic Dynamics

Abstract: Poincare's last geometric theorem asserts that an area preserving homeomorphism of the annulus twisting two boundaries admits two fixed points. The Arnold conjecture leading to the Floer theory is one way of generalizing the theorem. 

Another way of generalizing theorem by Birkhoff is to assume the twist is monotone from one boundary to the other, i.e. the so-called twist maps. Birkhoff, Morse, Hedlund obtained a complete characterization of the action minimizing orbits. The theory was revived by Aubry and Mather in the last 80s.

The motivation of Aubry was the transition from metal to insulator of a one-dimensional chain called the Frenkel-Kontorova model and his work left open a conjecture stating that when the ground states are uniformly hyperbolic, then the graph of the differential of the \alpha function is purely singularly continuous (called a complete devil staircase). In this talk, we explain our proof of Aubry's completeness conjecture. 

Related to this, we also explain another model called Hubbard-Wigner model exhibiting not only the phenomenon of complete devil staircase, but also admits an elegant analogue with Aubry-Mather theory. This model is considered as the thin torus limit of the fractional quantum Hall effects and the devil stair case reflects the fractional Hall conductance. In the talk, we also explain the passage from Aubry-Mather theory to the fractional quantum Hall effects.