Classification of hyperbolic discrete dynamics
Project/Area Number 
13640217

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  EHIME UNIVERSITY 
Principal Investigator 
HIRAIDE Koichi Ehime University, Faculty of Science Associate Professor, 理学部, 助教授 (50181136)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  dynamical systems / Anosov maps / 離散力学系 / 一様双曲型 / Anosov微分同相写像 / Axiom A微分同相写像 
Research Abstract 
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 wellknown, 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 codimensionone Anosov endomorphism of an arbitrary closed manifold. Suppose dimE^s = dim M  1. Then f is homotopically conjugate and inverselimit 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 codimensionone Anosov endomorphism of an arbitrary closed manifold. Suppose dim E^s = 1. Then f is homotopically conjugate and inverselimit conjugate to a hyperbolic infranilmanifold endomorphism of type dim E^s = 1. Futhermore, if f is special, then f is topologically conjugate to the hyperbolic infranilmanifold endomorphism. Theorem 3. The following (1), (2) and (3) hold ; (1) Two codimensionone Anosov endomorphisms are homotopically conjugate if and only if they are π_1conjugate. (2) Two codimensionone Anosov endomorphisms are inverselimit conjugate if and only if they are π_1conjugate up to finite index. (3) Two special codimensionone Anosov endomorphisms are topologically conjugate if and only if they are π_1conjugate. Less

Report
(3 results)
Research Products
(3 results)