One-dimensional basic sets in the three-sphere

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.