The realization of Smale solenoid type attractors in 3-manifolds

We study how to realize Smale solenoid type attractors in 3-manifolds. It is already known that we can restrict the 3-manifolds to lens spaces. We get all Smale solenoids realized in a given lens space through an inductive construction. We turn this around to address the question of how to decide whether a closed braid is a trivial knot in S3. For a diffeomorphism f of a 3-manifold M that realizes a Smale solenoid, it is natural to ask whether f−1 also realizes a Smale solenoid. We relate this question to exchangeable braids, and for some special positive case, we describe the relation between the two Smale solenoids of f and f−1.