| 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)
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| 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 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
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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
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