On a classification of axiom A diffeomorphisms with codimension one basic sets and isolated saddles
V. Medvedev, E. Zhuzhoma
Abstract
Let $M^n$, $n\geq 3$, be a closed orientable $n$-manifold and $\mathbb{D}_k(M^n;a,b,c)$ the set of axiom A diffeomorp\-hisms $f: M^n\to M^n$ satisfying the following conditions: (1) $f$ has $k\geq 1$ nontrivial basic sets each is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller, and other trivial basic sets which are $a$ sinks, $b$ sources, $c$ saddles; (2) the invariant manifolds of isolated saddles are intersected transversally. We classify the diffeomorphisms from $\mathbb{D}_k(M^n;a,b,c)$ up to the global conjugacy on non-wandering sets for the following subsets $\mathbb{S}_k(M^n;a,b,c), \mathbb{P}_k(M^n;0,0,1), \mathbb{M}_k(M^n;0,0,1)$ of $\mathbb{D}_k(M^n;a,b,c)$ where $\mathbb{S}_k(M^n;a,b,c)$ satisfies to the following conditions: ($1_{\mathbb{S}}$) every nontrivial basic set of any $f\in\mathbb{S}_k(M^n;a,b,c)$ is uniquely bunched, and there is at least one nontrivial attractor and at least one nontrivial repeller, i.e. $k\geq 2$; ($2_{\mathbb{S}}$) $c\geq 1$ and all isolated saddles have the same Morse index belonging to $\{1,n-1\}$. The subset $\mathbb{P}_k(M^n;0,0,1)\subset\mathbb{D}_k(M^n;0,0,1)$ satisfies to the following conditions: ($1_{\mathbb{P}}$) any boundary point of $f\in\mathbb{P}_k(M^n;0,0,1)$ is fixed; ($2_{\mathbb{P}}$) a unique isolated saddle has Morse index different from $\{1,n-1\}$. The subset $\mathbb{M}_k(M^n;0,0,1)\subset\mathbb{D}_k(M^n;0,0,1)$ satisfies to the following conditions: ($1_{\mathbb{M}}$) any boundary point of $f\in\mathbb{M}_k(M^n;0,0,1)$ is fixed; ($2_{\mathbb{M}}$) a unique isolated saddle has Morse index belonging to $\{1,n-1\}$. The classification is based on a description of topological structure of supporting manifolds $M^n$.