2000 Fiscal Year Final Research Report Summary
Topological theory of chaotic dynamics
| Project/Area Number |
09640116
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Ehime University |
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Principal Investigator |
HIRAIDE Koichi Ehime University, Paculty of Science Associate Frofessor, 理学部, 助教授 (50181136)
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| Project Period (FY) |
1997 – 2000
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| Keywords | dynamical systems / expansive maps / Anosov diffeomorphisms |
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| Research Abstract |
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^u
where ‖・‖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
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