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

Floer Homology, Nielsen Theory and Symplectic Zeta Functions

We describe a connection between symplectic Floer homology for symplectomorphisms of surface and Nielsen fixed point theory. A new zeta functions and asymptotic invariant of symplectic origin are defined. We show that special values of symplectic zeta functions are Reidemeister torsions.

New directions in Nielsen-Reidemeister theory

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

一个相对 Poincaré-Birkhoff 定理

A Relative Poincaré-Birkhoff theorem

本文证明适用于带 Legendrian 边界 Lagrangian 的相对 Poincaré-Birkhoff 定理，并利用局部 wrapped Floer 同调在扭转条件下得到任意长内部弦，服务于三体问题碰撞轨道研究。

$C^0$-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。

The growth rate of Floer homology and symplectic zeta function

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies …

一个关于某些$C^0$拉格朗日量的多维Birkhoff定理

A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

本文证明了一类新的$C^0$拉格朗日子集的多维Birkhoff定理，推广了环面扭转映射的经典结果，并首次将其扩展到仅连续对象，结合Floer理论、变分解和弱KAM理论。