On Chaotic Axiom A Diffeomorphisms with Codimension One Saddles and Basic Sets

We study Axiom A diffeomorphisms (in the sense of Smale) on a closed orientable $n$-manifold $M^n$, $n\geqslant 4$, whose nonwandering sets consist of codimension one orientable attractors, repellers, and isolated saddle periodic orbits with the same Morse index. Such systems combine chaotic dynamics on basic sets with regular dynamics near trivial periodic orbits. We prove that the supporting manifold $M^n$ is homeomorphic to a connected sum of several copies of the $n$-dimensional torus. Moreover, we obtain a complete classification of such systems up to global conjugacy on their nonwandering sets. The invariant of conjugacy is a finite collection of Anosov automorphisms of the torus, called a tube, defined up to equivalence under linear automorphisms with determinants of the same sign. This extends the classical Plykin – Grines classification of individual codimension one basic sets to systems with multiple coexisting attractors and repellers.