One-dimensional basic sets in the three-sphere

Gibbons, Joel C.

doi:10.1090/S0002-9947-1972-0292110-4

One-dimensional basic sets in the three-sphere
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by Joel C. Gibbons PDF

Trans. Amer. Math. Soc. 164 (1972), 163-178 Request permission

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.

References

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5
L. P. Neuwirth, Knot groups, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491–496. MR 196725, DOI 10.2307/2373203