A generalized Poincaré-Birkhoff theorem

$C^0$-哈密顿映射的庞加莱-伯克霍夫定理

A Poincaré--Birkhoff theorem for $C^0$-Hamiltonian maps

本研究利用Floer同调证明了庞加莱-伯克霍夫定理的高维版本及拉格朗日子流形的相对版本，旨在为圆形限制性三体问题中周期轨道和哈密顿弦的寻找提供理论支撑。

渐近正交哈密顿微分同胚的庞加莱-布克霍兹定理

A Poincaré-Birkhoff Theorem for Asymptotically Unitary Hamiltonian Diffeomorphisms

本文证明了在标准辛向量空间中，满足自然扭转条件的渐近线性非退化正交哈密顿微分同胚具有无限多个周期点。

A Poincaré-Birkhoff theorem for Hamiltonian flows on nonconvex domains

We present a higher-dimensional version of the Poincaré-Birkhoff theorem which applies to Poincaré time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an $N$-dimensional torus and the interior of a $(N-1)$-dimensional (not necessarily convex) embedded sphere; on the other hand, the classical boundary twist condition is replaced by an avoiding rays condition.

Poincaré-Birkhoff theorems in random dynamics

We propose a generalization of the Poincaré-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps that are random with respect to an ergodic probability measure. The classical theory is a particular instance of the random theory we propose.

A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link. This result admits a natural comparison to the Poincaré-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on $SO(3)$ and applies to geodesic flows of Finsler metrics on $S^2$.

Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.