权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Classification of hyperbolic discrete dynamics

双曲离散动力学的分类

基本信息

批准号：
13640217
负责人：
HIRAIDE Koichi
金额：
$ 1.98万
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：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

Let f : M → M be a regular C^1 map of a closed Riemannian manifold. We recall that f is an Anosov endomorphism if there are constants C > 0 and 0 < λ < 1 such that for any orbit (x_i) of f, i.e. f(x_i) = x_<i +I>, ∀_i ∈ Z, there is a splitting ∪_<i∈z>T_<x_i>M =E^s 【symmetry】 E^u = ∪_<i∈z>E^s_<x_i> 【symmetry】 E^u_<x_i>, which is left invariant by the derivative Df, such that for all n * 0 ‖Df^n(v)‖* Cλ^n‖v‖ if v ∈ E^s and ‖Df^n(v)‖ * C^<-1>λ^<-n> ||v|| if v ∈ E^u where || -‖ is the Riemannian metric. As is well-known, when (x_i) ≠ (y_i) and x_0 = y_0, we have E^u_<z_0> ≠ E^u_<y_0> in general. Hence, we will sometimes write E^u_<x_0> = E^u_<x_0>((x_i)). On the other hand, even if (x_i) ≠ (y_i), it follows that E^s_<x_0> = E^s_<y_0> whenever x_0=y_0, from which we have the stable bundle E^s = ∪_<x M>E^s_x, which is a continuous subbundle of the tangent bundle TM. We say that an Anosov endomorphism f : M →M is of codimension one if dim E^s = 1 or dimE^s = dim M - 1. We say that f is specia … More l if for orbits (x_i), (y_i) with x_0 = z_0, E^u_<x_0> = E^u_<y_0>. In this case we have the unstable bundle E^u = ∪_<z∈M> E^u_x^, which is also a continuous subbundle of TM. It is evident that if an Anosov e*morphism f : M → M is injective then f is special and it is an Anosov diffeomorphism, and that if E^s = 0, i.e. E^u = TM th** f is an expanding map, all of which form another class of special Anosov endomorphisms. In this study, the following theorems have been obtained ;Theorem 1. Let f : M → M be a codimension-one Anosov endomorphism of an arbitrary closed manifold. Suppose dimE^s = dim M - 1. Then f is homotopically conjugate and inverse-limit conjugate to a hyperbolic toral endomorphism of type dim E^s = dim M - 1. Futhermore, if f is special, then f is topologically conjugate to the hyperbolic toral endomorphism.Theorem 2. Let f : M → M be a codimension-one Anosov endomorphism of an arbitrary closed manifold. Suppose dim E^s = 1. Then f is homotopically conjugate and inverse-limit conjugate to a hyperbolic infra-nilmanifold endomorphism of type dim E^s = 1. Futhermore, if f is special, then f is topologically conjugate to the hyperbolic infra-nilmanifold endomorphism.Theorem 3. The following (1), (2) and (3) hold ; (1) Two codimension-one Anosov endomorphisms are homotopically conjugate if and only if they are π_1-conjugate. (2) Two codimension-one Anosov endomorphisms are inverse-limit conjugate if and only if they are π_1-conjugate up to finite index. (3) Two special codimension-one Anosov endomorphisms are topologically conjugate if and only if they are π_1-conjugate. Less

设f：M → M是闭黎曼流形上的正则C^1映射。我们记得f是Anosov自同态，如果存在常数C > 0和0 < λ < 1，使得对于f的任何轨道（x_i），即f（x_i）= x_<i +I>，λ_i ∈ Z，存在分裂λ_<i∈z>T_<x_i>M =E^s [对称性] E^u = λ_<i∈z>E^s_ [<x_i>对称性] E^u_<x_i>，它由导数Df保持不变，使得对于所有n * 0 λ_n <$Df^n（v）<$* Cλ^n <$v <$如果v ∈ E^s和λ_Df^n（v）<$* C^<-1>λ^<-n>||v||如果v ∈ E^u，其中||是黎曼度量。众所周知，当（x_i）≠（y_i）且x_0 = y_0时，我们一般有E^u_<z_0>≠ E^u_<y_0>。因此，我们有时会写E^u_<x_0>= E^u_<x_0>（（x_i））。另一方面，即使（x_i）<$（y_i），只要x_0=y_0，则E^s_<x_0>= E^s_<y_0>，由此我们得到稳定丛E^s =<$_<x M>E^s_x，它是切丛TM的连续子丛。我们说Anosov自同态f：M →M是余维1的，如果dimE ^s = 1或dimE^s = dimM- 1。我们说f是特殊的 ...更多信息 l如果对轨道（x_i），（y_i），其中x_0 = z_0，E^u_<x_0>= E^u_<y_0>.在这种情况下，我们有不稳定丛E^u = E^u_x^，它也是TM的一个连续子丛。显然，如果Anosov e* 态射f：M → M是单射的，则f是特殊的，并且它是Anosov自同态，如果E^s = 0，即E^u = TM th** f是扩张映射，所有这些都形成另一类特殊的Anosov自同态。在这项研究中，已经获得了以下定理：定理1。设f：M → M是任意闭流形的余维一Anosov自同态.假设dimE^s = dimM- 1。则f与dim E^s = dim M - 1型双曲环面自同态同伦共轭且逆极限共轭。进一步地，如果f是特殊的，则f与双曲环面自同态拓扑共轭。设f：M → M是任意闭流形的余维一Anosov自同态.假设dim E^s = 1。则f与dim E^s = 1型双曲次零流形自同态同伦共轭且逆极限共轭。进一步地，如果f是特殊的，则f拓扑共轭于双曲次零流形自同态。定理3。（1）余维1的Anosov自同态是同伦共轭的当且仅当它们是π_1-共轭的。(2)两个余维1的Anosov自同态是逆极限共轭的当且仅当它们在有限阶上是π 1-共轭的。(3)两个特殊的余维一Anosov自同态是拓扑共轭的当且仅当它们是π_1-共轭的。少