Smale solenoid attractors and affine Hirsch foliations

Abstract

The main purpose of this paper is to study north–south Smale solenoid diffeomorphisms on$3$-manifolds by using affine Hirsch foliations. A north–south Smale solenoid diffeomorphism$f$on a closed$3$-manifold$M$is a diffeomorphism whose non-wandering set is composed of a Smale solenoid attractor$\unicode[STIX]{x1D6EC}_{a}$and a Smale solenoid repeller$\unicode[STIX]{x1D6EC}_{r}$. The key observation is that a north–south Smale solenoid diffeomorphism$f$automatically induces two non-isotopically leaf-conjugate affine Hirsch foliations${\mathcal{H}}^{s}$and${\mathcal{H}}^{u}$on the orbit space of the wandering set of$f$(abbreviated to thewandering orbit spaceof$f$) by the stable and unstable manifolds of$\unicode[STIX]{x1D6EC}_{a}$and$\unicode[STIX]{x1D6EC}_{r}$, respectively. Under this viewpoint, we build some close relationships between north–south Smale solenoid diffeomorphisms and Hirsch manifolds (the closed$3$-manifolds admitting two non-isotopically leaf-conjugate affine Hirsch foliations).∙On the one hand, the union of the wandering orbit spaces is nearly in one-to-one correspondence with the union of Hirsch manifolds.∙On the other hand, surprisingly, the topology of the wandering orbit space (Hirsch manifold) is nearly a complete invariant of north–south Smale solenoid diffeomorphisms up to semi-global conjugacy.Moreover, as applications, we consider several more concrete questions. For instance, we prove that every diffeomorphism in many semi-global conjugacy classes of north–south Smale solenoid diffeomorphisms are not structurally stable.