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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



One-dimensional basic sets in the three-sphere

Author: Joel C. Gibbons
Journal: Trans. Amer. Math. Soc. 164 (1972), 163-178
MSC: Primary 58F10
MathSciNet review: 0292110
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Abstract: This paper is a continuation of Williams' classification of one-dimensional attracting sets of a diffeomorphism on a compact manifold [Topology 6 (1967)]. After defining the knot presentation of a solenoid in $ {S^3}$ and some knottheoretic preliminaries, we prove Theorem: If $ {\sum _1},{h_1}$ and $ {\sum _2},{h_2}$ are shift classes of oriented solenoids admitting elementary presentations K, $ K,{g_1}$ and K, $ K,{g_2}$, resp., where $ {g_1}^ \ast = {({g_2}^ \ast )^t}:{H_1}(K) \to {H_1}(K)$, there is an Anosov-Smale diffeomorphism f of $ {S^3}$ such that $ \Omega (f)$ consists of a source $ {\Lambda ^ - }$ and a sink $ {\Lambda ^ + }$ for which $ {\Lambda ^ + },f/{\Lambda ^ + }$ and $ {\Lambda ^ - },{f^{ - 1}}/{\Lambda ^ - }$ are conjugate, resp., to $ {\sum _1},{h_1}$ and $ {\sum _2},{h_2}$. (The author has proved [Proc. Amer. Math. Soc., to appear] that if f is an Anosov-Smale map of $ {S^3},\Omega (f)$ has dimension one, and contains no hyperbolic sets, then f has the above structure.) We also prove Theorem: there is a nonempty $ {C^1}$-open set $ {F_2}$ in the class of such diffeomorphisms for which $ K = {S^1}$ and $ {g_1} = {g_2}$ is the double covering such that each f in $ {F_2}$ defines a loop t in $ {S^3}$, stable up to $ {C^1}$ perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.

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Keywords: Generalized solenoid, structural instability, knot factorization
Article copyright: © Copyright 1972 American Mathematical Society

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