单连通区域上全纯自同构的拓扑共轭分类

发布时间：2018-09-10 16:20

【摘要】：本文主要介绍了单连通区域上全纯自同构的拓扑共轭分类.我们说两个变换f:X→X和g:Y→Y是拓扑共轭的,如果存在一个同胚h:X→Y使得h(?)f=g(?)h,这里(?)是映射的复合.单连通区域主要包括:复平面,扩充复平面,单位圆盘和上半平面.关于复平面上全纯自同构的拓扑共轭分类问题,Budnitska得到了更一般的结论,即有限维向量空间上仿射算子的拓扑共轭分类.具体地讲,如果仿射算子f(x)=Ax+b有一个不动点,那么f拓扑共轭于它的线性部分A;如果仿射算子f:U→U无不动点,我们证明f拓扑共轭于一个仿射算子g:U→U,这里U是g的不变子空间V和W的正交直和,g在V上的限制g|V是一个拥有如下形式的V的标准正交基底的仿射算子(x1,x2,...,xn)→(x1+1,x2,...,xn-1,εxn),ε=±1,它是由f唯一确定的,g在W上的限制g|W是一个通过幂零Jordan矩阵给出的W的标准正交基底的线性算子,是由f唯一确定的.对于扩充复平面上的分式线性变换的拓扑共轭分类问题,Rybalkina和S ergeichuk已经给出了相应的结果.我们在此整理并总结了复平面和扩充复平面上全纯自同构的拓扑共轭分类,关于单位圆盘和上半平面上全纯自同构的拓扑共轭之前没有得到完全的分类,本文将就这一问题给出答案.我们通过旋转理论及构造同胚的方法,证明了:上半平面上(单位圆盘)所有无不动点的全纯自同构之间都是拓扑共轭的;两个有不动点的全纯自同构f和g是拓扑共轭的当且仅当ρ(f)=±ρ(g),modZ;无不动点的全纯自同构与有不动点的全纯自同构之间是不拓扑共轭的.
[Abstract]:In this paper, the topological conjugate classification of Holomorphic automorphism on a simple connected domain is introduced. We say that the two transformations f: X X and g: y y are topological conjugate, if there is a homomorphism h: X y such that h (?) FG (?) h, here (?) Is a composite of maps. Simple connected regions mainly include: complex plane, extended complex plane, unit disk and upper half plane. On topological conjugate classification of Holomorphic automorphism on complex plane Budnitska obtained a more general conclusion that is topological conjugate classification of affine operators on finite dimensional vector space. In particular, if the affine operator f (x) n Ax b has a fixed point, then f topology conjugates to its linear part A, and if the affine operator f: U U U has no fixed point, We prove that f topology is conjugate to an affine operator g: U, where U is the orthogonal direct sum of the invariant subspace V and W of g on V, g V is an affine operator with a standard orthonormal base of V in the following form. N (x 11 1 + x 2n 1, 蔚 xn), 蔚 = 卤1), which is the limit g W on W determined only by f is a linear operator of the standard orthogonal base of W given by nilpotent Jordan matrix. Is determined only by f. For topological conjugate classification of fractional linear transformations on extended complex plane, the corresponding results have been given for Rybalkina and S ergeichuk. The topological conjugate classification of Holomorphic automorphism on complex plane and extended complex plane is summarized. The topological conjugate of Holomorphic automorphism on unit disk and upper half plane has not been completely classified before. This article will give the answer to this question. By means of rotation theory and the method of constructing homeomorphism, we prove that all Holomorphic automorphisms without fixed points on the upper half plane (unit disk) are topological conjugate; Two Holomorphic automorphisms f and g with fixed points are topological conjugate if and only if 蟻 (f) = 卤蟻 (g) mod Z, and between Holomorphic automorphism without fixed point and Holomorphic automorphism with fixed point is not topological conjugate.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189.11

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