On the Topological Structure of Manifolds Supporting Axiom A Systems

Regular and Chaotic Dynamics. 2022. Vol. 27. No. 6. P. 613–628.

Vyacheslav Z. Grines, Vladislav S. Medvedev, Evgeny V. Zhuzhoma

Let M^n, , n⩾3, be a closed orientable n-manifold and G(M^n) be the set of A-diffeomorphisms f:M^n→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 S^n or the connected sum of k_f⩾0 copies of the torus T^n, η_f⩾0 copies of S^(n−1)×S^1,  and l_f⩾0 simply connected manifolds N^n_1,…,N^n_{lf} which are not homeomorphic to the sphere. Here k_f⩾0 is the number of connected components of all nontrivial basic sets, η_f=κ_f/2−(k_f+ν_f−μ_f+2)/2, is the number of bunches of all nontrivial basic sets, μ_f⩾0 is the number of sinks and sources, ν_f⩾0 is the number of isolated saddle periodic points with Morse index 1  or n−1, 0⩽l_f⩽λ_{f0}, λf⩾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.

Keywords: gradient-like diffeomorphism, A-diffeomorphism, structure of non-wandering set and topology of ambient manifold

Publication based on the results of: Теория динамических систем и ее приложения (2021)