One-dimensional basic sets in the three-sphere
Joel C. Gibbons
Trans. Amer. Math. Soc. 164 (1972), 163-178
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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 and some knottheoretic preliminaries, we prove Theorem: If and are shift classes of oriented solenoids admitting elementary presentations K, and K, , resp., where , there is an Anosov-Smale diffeomorphism f of such that consists of a source and a sink for which and are conjugate, resp., to and . (The author has proved [Proc. Amer. Math. Soc., to appear] that if f is an Anosov-Smale map of has dimension one, and contains no hyperbolic sets, then f has the above structure.) We also prove Theorem: there is a nonempty -open set in the class of such diffeomorphisms for which and is the double covering such that each f in defines a loop t in , stable up to perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.
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