Topological theory of chaotic dynamics

混沌动力学拓扑理论

• 批准号: 09640116
• 负责人: HIRAIDE Koichi
• 金额: $ 0.96万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640116/
• 关键词: dynamical systems; expansive maps; Anosov diffeomorphisms

Let f : M → M be a diffeomorphism of a closed Riemannian manifold. We recall that f is an Anosov diffeomorphism if there are constants c > 0 and 0 < λ < 1, and a continuous splitting TM = E^s 【symmetry】 E^u of the tangent bundle, which is left invariant by the derivative D f, such that for all n 【greater than or equal】 0‖Df^n(υ)‖【less than or equal】 cλ^n‖υ‖if υ ∈ E^s, and ‖Df^<-n>(υ)‖【less than or equal】 cλ^n‖υ‖ if υ ∈ E^uwhere ‖・‖is the Riemannian metric. An Anosov diffeomorphism f is said to be of codimension one if dim E^s = 1 or dim E^u = 1. The following well-known theorem is the conclusion of Theorems 2 and 3 below, which were proved by J.Franks and S.E.Newhouse respectively.Theorem 1. If f : M → M is a codimension one Anosov diffeomorphism, then f is topologically conjugate to a hyperbolic toral automorphism.This research gave simple proofs of Theorems 2 and 3.Theorem 2 (Franks). If an Anosov diffeomorphism f : M → M is of codimension one and the nonwandering set Ω(f) coincides … More with the whole space M, then f is topologically conjugae to a hyperbolic toral automorphism.Theorem 3 (Newhouse). If an Anosov diffeomorphism f : M → M is of codimension one, then Ω ( f) = M.In addition, this research classified codimension one Anosov endomorphisms by applying the ides in the proofs of the above theorems.Let f : M → M be a C^r diffeomorphism of a closed manifold, 0 【less than or equal】 r 【less than or equal】∞, and let m^0 be a fixed point of f. A closed manifold is a compact connected manifold without boundary and supposed to have a smooth structure if r 【greater than or equal】 1. By a C^0 diffeomorphism will be meant a homeomorphism of a topological manifold. We say that f is a π_1-diffeomorphism (with base point m_0) if for a homeomorphism g : K → K of a compact CW complex with fixed point k_0 and for a continuous map h' : K → M with h'(k_0) = m_0 if f_* o h'_* = h'_* o g_* on the fundamental groups, then there is a unique continuous map h : K → M, free homotopic to h', with h(k_0) = m_0 such that f o h = h o g. This notion was introduced by Franks, in 1970, in connection with the problem of classifying all Anosov diffeomorphisms of closed manifolds up to topological conjugacy Franks proved that two π_1 diffeomorphisms f : M → M and g : N → N are topologically conjugate if and only if the induced automorphisms f_* and g_* on the fundamental groups are algebraically conjugate, and that every hyperbolic infra-nilmanifold automorphism, which is an extension of hyperbolic toral automorphisms, is a π_1-diffeomorphism.This research gave an answer to the problem, posed by Franks, of classifying all π_1-diffeomorphisms up to topological conjugacy.Theorem 4. A π_1-diffeomorphism of an arbitrary closed manifold is topologically conjugate to a hyperbolic infra nilmanifold automorphism. Less

设f：M → M是闭黎曼流形的一个单同态.我们记得f是一个Anosov同态，如果有常数c &gt; 0和0 &lt; λ &lt; 1，以及切丛的一个连续分裂TM = E^s [对称性] E^u，它被导数D f左不变，使得对于所有n [大于或等于] 0 &lt;$Df^n（n）&lt;$[小于或等于] cλ^n &lt;$如果&lt;$∈ E^s，和&lt;$Df^<-n>（n）&lt;$[小于或等于] c λ ^n &lt;$如果&lt;$∈ E^u，其中&lt;$·&lt;$是黎曼度量。如果dim E^s = 1或dim E^u = 1，则称Anosov同态f具有余维数1。下面的公知定理是下面的定理2和定理3的结论，它们分别由J.Franks和S. E.纽豪斯证明。如果f：M → M是余维数为1的Anosov自同构，则f与双曲型环面自同构拓扑共轭.本文给出了定理2和定理3的简单证明.若Anosov同态f：M → M是余维1的，且非游荡集Ω（f）重合 ...更多信息 定理3（纽豪斯）.若Anosov自同态f：M → M是余维数为1的，则Ω（f）= M.另外，利用上述定理证明中的定理，对余维数为1的Anosov自同态进行了分类.设f：M → M是闭流形的C^r自同态，0 [小于等于] r [小于等于]∞，m^0是f的不动点.一个闭流形是一个没有边界的紧致连通流形，如果r [大于或等于] 1，则它被认为具有光滑结构。C^0同胚是指拓扑流形的同胚。我们称f是π_1-同态（以m_0为基点）如果对具有不动点k_0的紧CW复形的同胚g：K → K和对具有h '（k_0）= m_0的连续映射h'：K → M，如果f_* oh '_* = h'_* og_*，则在基本群上存在唯一的连续映射h：K → M，h '的自由同伦，其中h（k_0）= m_0使得fohg.这个概念是由Franks在1970年引入的，在将闭流形的所有Anosov同态分类到拓扑共轭的问题中，Franks证明了两个π_1同态f：M → M和g：N → N是拓扑共轭的当且仅当基本群上的诱导自同构f * 和g * 是代数共轭的，证明了作为双曲环自同构的推广的双曲次幂零流形自同构是π_1-π-同态，从而回答了Franks提出的将π_1-π-同态分类到拓扑共轭的问题。任意闭流形的π_1-π-同态拓扑共轭于双曲下零流形的自同构。少

项目成果

Koichi Hiraide: "A simple proof of the Franks-Newhouse theorem on codimension-one Anosov diffeomorphisms"Ergod.Th.& Dynam.Sys.. 21. 1-6 (2001)

Koichi Hiraide：“关于余维一阿诺索夫微分同胚的弗兰克斯-纽豪斯定理的简单证明”Ergod.Th。