Erratum to "Generalizations of the Poincaré-Birkhoff Theorem"

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. The case of complex dynamics is investigated, too.

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.

A topological version of the Poincaré-Birkhoff theorem with two fixed points

The main result of this paper gives a topological property satisfied by any homeomorphism of the annulus $\mathbb{A}=\mathbb{S}^1 \times [-1,1]$ isotopic to the identity and with at most one fixed point. This generalizes the classical Poincaré-Birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism of $\mathbb{A}$ with the usual boundary twist condition. We also have two corollaries of this result. The first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism $h$ has a lift $H$ to the strip $…

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.

$C^0$-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。