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
{S^3}
and some knottheoretic preliminaries, we prove Theorem: If
∑
1
,
h
1
{\sum _1},{h_1}
and
∑
2
,
h
2
{\sum _2},{h_2}
are shift classes of oriented solenoids admitting elementary presentations K,
K
,
g
1
K,{g_1}
and K,
K
,
g
2
K,{g_2}
, resp., where
g
1
∗
=
(
g
2
∗
)
t
:
H
1
(
K
)
→
H
1
(
K
)
{g_1}^ \ast = {({g_2}^ \ast )^t}:{H_1}(K) \to {H_1}(K)
, there is an Anosov-Smale diffeomorphism f of
S
3
{S^3}
such that
Ω
(
f
)
\Omega (f)
consists of a source
Λ
−
{\Lambda ^ - }
and a sink
Λ
+
{\Lambda ^ + }
for which
Λ
+
,
f
/
Λ
+
{\Lambda ^ + },f/{\Lambda ^ + }
and
Λ
−
,
f
−
1
/
Λ
−
{\Lambda ^ - },{f^{ - 1}}/{\Lambda ^ - }
are conjugate, resp., to
∑
1
,
h
1
{\sum _1},{h_1}
and
∑
2
,
h
2
{\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
,
Ω
(
f
)
{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
{C^1}
-open set
F
2
{F_2}
in the class of such diffeomorphisms for which
K
=
S
1
K = {S^1}
and
g
1
=
g
2
{g_1} = {g_2}
is the double covering such that each f in
F
2
{F_2}
defines a loop t in
S
3
{S^3}
, stable up to
C
1
{C^1}
perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.