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.
Smale, Differentiable dynamical
systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014
(37 #3598), http://dx.doi.org/10.1090/S0002-9904-1967-11798-1
F. Williams, One-dimensional non-wandering sets, Topology
6 (1967), 473–487. MR 0217808
P. Neuwirth, Knot groups, Annals of Mathematics Studies, No.
56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
Smale, Structurally stable systems are not dense, Amer. J.
Math. 88 (1966), 491–496. MR 0196725
- S. Smale, Differential dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37 #3598. MR 0228014 (37:3598)
- R. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473-487. MR 36 #897. MR 0217808 (36:897)
- L. Neuwirth, Knot groups, Ann of Math. Studies, no. 56, Princeton Univ. Press, Princeton, N. J., 1965. MR 31 #734. MR 0176462 (31:734)
- S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491-496. MR 33 #4911. MR 0196725 (33:4911)
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