This is a note taken from Bowen, Periodic Orbits for Hyperbolic Flows, American Journal of Mathematics (1972), 1–30.

We start with Anosov dichotomy:

Let be a transitive Anosov flow. Then

1. either it is mixing: then strong stable and strong unstable manifold everywhere dense on

2. or it is a suspension: choose the closure of a non-dense stable manifold as a cross-section and the induced roof-function is constant.

Anosov proved this in volume-preserving case and Plante proved it for the general case (Anosov flows, 1972).

Let be a closed manifold and be a flow. Let be a closed, invariant subset without fixed points, that is, the vector field of does not admits zeros on . Then is said to be a basic set of if

– is (topologically) transitive and hyperbolic,

– close orbits are dense in

– is isolated: for some open neighborhood of .

The last one is equivalent to the local product structure.

Theorem 3.2 in [B]. There are three mutually exclusive types:

1. consists of a single closed orbit of ;

2. the strong stable manifold is dense in for for each ;

3. is the constant suspension of a Axiom A homeomorphism.

As remarked by Bowen, this is first proved by Anosov for volume-preserving Anosov flow, by Plante for general Anosov flow. We should view the following as a proof for Anosov flow case for first reading, and then take the induced topology on for basic sets.