Classification of hyperbolic discrete dynamics

设f: M→M是一个闭黎曼流形的正则C^1映射。我们回想一下,f是一个Anosov自同态如果有常数C > 0, 0 <λ< 1,这样对任何轨道(x_i)的f,即f (x_i) =间的<我+ >,∀_i∈Z,有一个分裂∪_ <我∈Z > T_ < x_i > M = E ^ s【对称】E ^ u =∪_ <我∈Z > E ^ s_ < x_i >【对称】E ^ u_ < x_i >,由导数Df左不变,这样所有n * 0为Df ^ n (v)为* Cλ^ n为v为如果v∈E ^年代为Df ^ n (v)为* C ^ < 1 >λ^ < - n > | | v | |如果u v∈E ^ | | -为黎曼度量。众所周知，当(x_i)≠(y_i)且x_0 = y_0时，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)，当x_0=y_0时，E^s_<x_0> = E^s_<y_0>，由此我们得到稳定束E^s =∪_<x M>E^s_x，它是切束TM的连续子束。我们说，当dimE^s = 1或dimE^s = dim M - 1时，Anosov自同态f: M→M的余维为1。我们说f是特殊的，对于轨道（x_i）， （y_i），当x_0 = z_0, E^u_<x_0> = E^u_<y_0>。在这种情况下，我们得到不稳定束E^u =∪_<z∈M> E^u_x^，它也是TM的连续子束。显然，如果一个Anosov e*态射f: M→M是内射，则f是一个特殊的Anosov微分同态；如果e ^s = 0，即e ^u = TM，则** f是一个展开映射，这些都形成了另一类特殊的Anosov自同态。本研究得到以下定理：定理1。设f: M→M是任意闭流形的一个协维1的Anosov自同态。假设dimE^s = dim M - 1。那么f是同伦共轭的，并且逆极限共轭于一个类型为dim E^s = dim M - 1的双曲型总自同态。进一步，如果f是特殊的，则f拓扑共轭于双曲总自同态。定理2。设f: M→M是任意闭流形的一个协维1的Anosov自同态。假设dim E^s = 1。然后f是同伦共轭的，并且逆极限共轭于一个dim E^s = 1型的双曲次零流形自同态。进一步，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)、(2)、(3)成立；(1)两个余维一Anosov自同态当且仅当π共轭时为同伦共轭。(2)两个协维一Anosov自同态当且仅当π共轭至有限指数时为反极限共轭。(3)两个特殊的余维一Anosov自同态当且仅当π共轭时是拓扑共轭的。少

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