Topological theory of chaotic dynamics

令 f : M → M 为闭黎曼流形的微分同胚。我们回想一下，如果存在常数 c > 0 且 0 < λ < 1，并且切束的连续分裂 TM = E^s [对称性] E^u，则 f 是阿诺索夫微分同胚，它通过导数 D f 保持不变，使得对于所有 n [大于或等于] 0‖Df^n(υ)‖【小于或等于】cλ^n‖υ‖如果 υ ∈ E^s，并且‖Df^<-n>(υ)‖【小于或等于】 cλ^n‖υ‖ 如果 υ ∈ E^u 其中 ‖・‖是黎曼度量。如果dim E^s = 1 或dim E^u = 1，则阿诺索夫微分同胚 f 被称为余维一。以下众所周知的定理是下面定理 2 和定理 3 的结论，分别由 J.Franks 和 S.E.Newhouse 证明。定理 1. 如果 f : M → M 是余维一阿诺索夫微分同胚，则 f 在拓扑上共轭于双曲环图自同构。这项研究给出了定理 2 和 3 的简单证明。定理 2 (Franks)。如果阿诺索夫微分同胚 f : M → M 具有余维 1 并且非游走集 Ω(f) 与整个空间 M 重合，则 f 在拓扑上与双曲环自同构共轭。定理 3 (Newhouse)。若阿诺索夫微分同胚 f : M → M 为余维一，则 Ω ( f) = M。此外，本研究应用上述定理证明中的 ides，对余维一阿诺索夫自同胚进行分类。设 f : M → M 为闭流形的 C^r 微分同胚，0 【小于等于】 r 【小于等于】∞，设 m^0 为 f 的不动点。闭流形是一个没有边界的紧连通流形，如果 r 【大于或等于】1，则应该具有光滑结构。C^0 微分同胚意味着拓扑流形的同胚。我们说 f 是 π_1-微分同胚（基点为 m_0），如果对于具有不动点 k_0 的紧致 CW 复形的同态 g : K → K 以及对于连续映射 h' : K → M 且 h'(k_0) = m_0 如果基本群上的 f_* o h'_* = h'_* o g_* ，则存在唯一的连续映射 h : K → M，与 h' 自由同伦，其中 h(k_0) = m_0 使得 f o h = h o g。这一概念由 Franks 于 1970 年提出，结合将闭流形的所有阿诺索夫微分同胚分类到拓扑共轭的问题 Franks 证明了两个 π_1 微分同胚 f : M → M 和 g : N → N 是拓扑共轭的，当且仅当基本群上的导出自同构 f_* 和 g_* 是代数共轭时，并且每个双曲下零流形自同构是双曲环自同构的扩展，是一个 π_1-微分同胚。这项研究回答了 Franks 提出的将所有 π_1-微分同胚分类为拓扑共轭的问题。定理 4。任意闭流形的 π_1-微分同胚在拓扑上共轭于双曲下尼尔流形自同构。较少的

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