\({\mathcal X}^*\) plus axiom A does non imply no-cycle
Let \(M\) be a smooth closed maifold, \(\text{Diff}^1 (M)\) the set of diffeomorphisms and \({\mathcal X}^1 (M)\) the set of \(C^1\) vector fields of \(M\), both endowed with the \(C^1\) topology. Denote by \({\mathcal F}^* (M)\) the set of \(f\in \text{Diff}^1 (M)\) with the property: \(f\) has a \(C^1\) neighborhood \(U\) in \(\text{Diff}^1 (M)\) such that every \(g\in U\) has no non-hyperbolic periodic points, and by \({\mathcal X}^* (M)\) the set of \(X\in {\mathcal X}^1 (M)\) such that \(X\) has a \(C^1\) neighborhood \(U\) in \({\mathcal X}^1 (M)\) and every \(Y\in U\) has neither non-hyperbolic periodic orbits, nor non-hyperbolic singularities. It is known [cf. \textit{J. Palis}, Publ. Math., Inst. Hautes Étud. Sci. 66, 211-215 (1988; Zbl 0648.58019) and \textit{R. Mañe}, Ann. Math., II. Ser. 116, 503-540 (1982; Zbl 0511.58029)]\ that if \(f\in {\mathcal F}^* (M)\) then \(f\) satisfies Axiom A and the no-cycle condition. For vector fields the situation is quite different. From \textit{J. Guckenheimer} [Appl. Math. Ser. 19, 165-178 (1976; Zbl 0346.58007)]\ it follows that \(X\in {\mathcal X}^* (M)\) does not imply Axiom A. In the paper it is shown that \(X\in {\mathcal X}^* (M)\) and \(X\) satisfies Axiom A do not imply the no-cycle condition. More precisely, the following result holds. Theorem: There is a vector field \(Z\) on the sphere \(S^2\) such that \(Z\in {\mathcal X}^* (S^2)\), \(Z\) satisfies Axiom A and does not satisfy the no-cycle condition.
scientific article; zbMATH DE number 784945