Topological theory of chaotic dynamics

让f ： M → M是一种封闭的利曼尼亚式的差异形态。我们认为，如果f是一个常数c > 0和0 < λ < 1，并且连续的分裂TM = E^s [对称性] E^u的切线捆绑，它是衍生D f的左变量，因此对于所有n [大于或等于] 0 μ Df^n（）[小于或等于] c μ n如果E^s，and\Df^\-n>（\）\[less than or equal] c\^n\\if\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。下面已知的定理是定理2和3的集合，J. Franks和S. E. Newhouse对此提出的。如果f ：M → M是Anosov差异化的一个编码，然后F是一个超自然的超自然现象。本研究提供了理论2和3的简单证明。理论2 （弗兰克）。如果Anosov Diffeomorphism f ： M → M是一个编码和一个非漫游集Ω（f）共因 ... More 在整个空间M中，然后f在拓扑上是一个超自然的超自然现象。理论3 （Newhouse）。如果Anosov差异化f ： M → M是编码一个，然后Ω （ f） = M.添加，本研究的分类编码一个Anosov内形态化应用于上理论的证明。让f ： M → M是一个封闭的歧管， 0 [小于或等于] r [小于或等于]小于或等于] ∞，让m^0是f的固定点。一个封闭的manifold是一个没有边界和支持有一个光滑的结构的紧凑连接manifold，如果r [大于或等于] 1。通过C^0的差异性将意味着一种拓扑的同质性。我们说f是一个π_1-差别形态（与基点m_0）如果一个同源性g ： K →一个固定点k_0的紧凑CW复合体的K，而一个连续图h' ： K → M，带有h'（k_0） = m_0如果f_* o h'_* = h'_* o g_*，那么有一个独特的连续图h ：K → M，与h（k_0） = m_0这样f o h = h o g。这一运动是由Franks于1970年提出的，在与所有封闭的歧管类问题的关联中，Franks提出了两个π_1差管f ： M → M和g ：N → N是在拓扑上被混淆的，如果并且只有在基础群体上被诱导的自动形态学f_* 和g_* 是在代数上被混淆的情况下，而且，每一种超线性非线性自动化，它是超线性toral自动化的扩展，是π_1-差异形态化，这一研究对这个问题提出了一个答案，由Franks提出，将所有π_1-差异形态化到拓扑并发症。理论4。A π_1-一个封闭的曲线形的差异在拓扑上与一个超线性的非线性自动形态相结合。Less（低）

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