On the Topological Structure of Manifolds Supporting Axiom A Systems

Let $M^n$, $n\geqslant 3$, be a closed orientable $n$-manifold and $\mathbb{G}(M^n)$ the set of $\mathrm{A}$-diffeomorphisms $f: M^n\to M^n$ whose nonwandering set satisfies the following conditions: $(1)$ each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller; $(2)$ the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only one-dimensional separatrices of other isolated periodic orbits. We prove that the ambient manifold $M^n$ is homeomorphic to either the sphere $\mathbb S^n$ or the connected sum of $k_f \geqslant 0$ copies of the torus $\mathbb T^n$, $\eta_f\geqslant 0$ copies of $\mathbb S^{n-1}\times \mathbb S^1$ and $l_f\geqslant 0$ simply connected manifolds $N^n_1, \dots, N^n_{l_f}$ which are not homeomorphic to the sphere. Here $k_f\geqslant 0$ is the number of connected components of all nontrivial basic sets, $\eta_{f}=\frac{\kappa_f}{2} -k_f+\frac{\nu_f - \mu_f +2}{2},$ $ \kappa_f\geqslant 0$ is the number of bunches of all nontrivial basic sets, $\mu_f\geqslant 0$ is the number of sinks and sources, $\nu_f\geqslant 0$ is the number of isolated saddle periodic points with Morse index $1$ or $n-1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n-1,n\}$ of diffeomorphism $f$. Similar statements hold for gradient-like flows on $M^n$. In this case there are no nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient conditions for the existence of heteroclinic intersections and periodic trajectories for Morse – Smale flows.