On Diffeomorphisms with Orientable Codimension 1 Basic Sets and an Isolated Saddle

We consider the class $$\mathbb{G}_k^{\text{diff}}(M^n;0,0,1)$$ of diffeomorphisms $$f \colon\, M^n\to M^n$$ of a closed orientable $$n$$ -manifold $$M^n$$ , $$n\geq 3$$ , that satisfy Smale’s axiom A whose nonwandering set $$\mathrm{NW}(f)$$ consists of the following basic sets: (a) $$k\geq 1$$ nontrivial basic sets each of which is either an orientable connected expanding codimension $$1$$ attractor or an orientable connected contracting codimension $$1$$ repeller; (b) exactly one trivial basic set, an isolated saddle, whose separatrices do not intersect. For diffeomorphisms in $$\mathbb{G}_k^{\text{diff}}(M^n;0,0,1)$$ , we construct a certain equipped graph that gives a complete global conjugacy invariant on their nonwandering sets. We also describe the topological structure of the supporting manifolds $$M^n$$ for diffeomorphisms in the class $$\mathbb{G}_k^{\text{diff}}(M^n;0,0,1)$$ , $$n\geq 3$$ , $$n\neq 4$$ , $$k\geq 2$$ .