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

We consider the class $\mathbb {G}_k^{\textup {diff}}(M^n;0,0,1)$ of diffeomorphisms $f: M^n\to M^n$ 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 $\mathbb {G}_k^{\textup {diff}}(M^n;0,0,1)$, 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 $\mathbb {G}_k^{\textup {diff}}(M^n;0,0,1)$, $n\geq 3$, $n\neq 4$, $k\geq 2$.