Motivated by Morse theory and Smale's work in dynamics, as well as our study of cheirality, the following questions are studied and answered, in joint work with Boju Jiang and Yi Ni:
(1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as attractor?
(2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids?
The result presents some intrinsic symmetries for a class of 3-manifolds.
Furthermore, the simplest example in (2) involves the discovery of a non-fibred knot whose infinite cyclic covering embeds into the 3-sphere. The existence of such an example was asked by Stallings and Edwards 20 years ago.
We also give some obstruction to embedding the infinite cyclic coverings of compact 3-manifolds into compact 3-manifolds.