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 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*.

**[1]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, 10.1090/S0002-9904-1967-11798-1**[2]**R. F. Williams,*One-dimensional non-wandering sets*, Topology**6**(1967), 473–487. MR**0217808****[3]**L. P. Neuwirth,*Knot groups*, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR**0176462****[4]**S. Smale,*Structurally stable systems are not dense*, Amer. J. Math.**88**(1966), 491–496. MR**0196725**

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0292110-4

Keywords:
Generalized solenoid,
structural instability,
knot factorization

Article copyright:
© Copyright 1972
American Mathematical Society