Abstract
Abstract
Let f
t
be a flow satisfying Smale’s Axiom A (in short, A-flow) on a closed orientable three-manifold M
3, and Ω a two-dimensional basic set of f
t
. First, we prove that Ω is either an expanding attractor or contracting repeller. Next, one considers an A-flow f
t
with a two-dimensional non-mixing attractor Λ
a
. We construct a casing M(Λ
a
) of Λ
a
that is a special compactification of the basin of Λ
a
by a collection of circles L(Λ
a
) = {l
1, …, l
k
} such that M(Λ
a
) is a closed three-manifold and L(Λ
a
) is a fibre link in M(Λ
a
). In addition, f
t
is extended on M(Λ
a
) to a nonsingular structurally stable flow with the non-wandering set consisting of the attractor Λ
a
and the repelling periodic trajectories l
1, …, l
k
. We show that if a closed orientable three-manifold M
3 has a fibred link L = {l
1, …, l
k
} then M
3 admits an A-flow f
t
with the non-wandering set containing a two-dimensional non-mixing attractor and the repelling isolated periodic trajectories l
1, …, l
k
. This allows us to prove that any closed orientable n-manifold, n ⩾ 3, admits an A-flow with a two-dimensional attractor. We prove that the pair
M
(
Λ
a
)
;
L
(
Λ
a
)
consisting of the casing M(Λ
a
) and the corresponding fibre link L(Λ
a
) is an invariant of conjugacy of the restriction
f
t
|
W
s
(
Λ
a
)
of the flow f
t
on the basin of the attractor Λ
a
.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Reference51 articles.
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2. Geodesic flows on closed Riemannian manifolds of negative curvature;Anosov;Trudy Mat. Inst. Steklov.,1967
3. On a class of invariants sets of smooth dynamical systems;Anosov,1970
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