C 1 stable maps: examples without saddles.

The article deals with C 1 structurally stable maps on compact manifolds of dimension greater than two. The map f : M → M is an Axiom A if its nonwandering set is hyperbolic and the set of periodic points of f is dense in Ω(f). An attractor Λ of f : M → M is called topologically simple if there exists a neighborhood U of Λ with the following properties: U is contained in the basin of attraction of Λ; the closure of f(U) is contained in U; the restriction of f to U is injective; each connected component of U intersects Λ; and for each closed curve γ in U there exists a closed curve γ ′ in f(U) such that γ and γ ′ are homotopic and the corresponding homotopy is contained in U. The following statements are proved: (Theorem 1) If f ∈ C 1 (M), is an Axiom A map without critical points, and every basic piece is either expanding or attracting, then f is C 1 structurally stable; (Theorem 2) If M is a manifold admitting an expanding map and embedded into some sphere S, then there exists a noninvertible Axiom A map f in C 1 (M × S) whose nonwandering set is the union of an expanding set and a nonperiodic attractor; moreover f has no critical points and, due to Theorem 1, C 1 structurally stable; (Theorem 3) If M is connected and f ∈ C 1 (M) is nonivertible Axiom A map without critical points with a topologically simple attractor Λ, then (a) the restriction of f to B0, the immediate basin of attraction of Λ, is injective, and (b) in the boundary of B0 there are nonwandering points that do not belong to an expanding basic piece.