On reversible maps and symmetric periodic points
On reversible maps and symmetric periodic points
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DOI：
10.1017/etds.2016.71
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发表时间：
2014-10
影响因子：
0.9
通讯作者：
Jungsoo Kang
中科院分区：
文献类型：
--
作者：
Jungsoo Kang
在可逆动力系统中，了解对称性是非常重要的。本文的目的是研究平面区域上可逆映射的对称周期点在反射下的不变性。本文将关于环上保面积映射周期点个数的二分性的Franks定理推广到保面积可逆映射的对称周期点。有趣的是，即使一个非对称周期点也保证了无穷多个对称周期点。证明了保面积可逆映射的对称奇周期点与恒等式同构的一个类似命题，该命题可应用于具有双重对称性的动力系统。我们的方法是简单的，基本的，远离弗兰克斯的证明。我们还证明了可逆映射有对称不动点当且仅当它是一个扭曲映射，它推广了Poincaré-Birkhoff意义上的封闭环上的边界扭曲条件。简要讨论了对称周期轨道在二自由度可逆动力系统中的应用。
In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.