Topological theory of chaotic dynamics
Project/Area Number  09640116 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Ehime University 
Principal Investigator 
HIRAIDE Koichi Ehime University, Paculty of Science Associate Frofessor, 理学部, 助教授 (50181136)

Project Period (FY) 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  dynamical systems / expansive maps / Anosov diffeomorphisms 
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 wellknown 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
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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_1diffeomorphism (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 infranilmanifold automorphism, which is an extension of hyperbolic toral automorphisms, is a π_1diffeomorphism. This research gave an answer to the problem, posed by Franks, of classifying all π_1diffeomorphisms up to topological conjugacy. Theorem 4. A π_1diffeomorphism of an arbitrary closed manifold is topologically conjugate to a hyperbolic infra nilmanifold automorphism. Less

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Research Products
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