\(C^1\)-maps having hyperbolic periodic points (Q2717759)

Let \(M\) be a closed \(C^{\infty}\)-manifold, and denote by \(C^{1}(M)\) the space of all \(C^{1}\)-maps \(f:M\to M\) endowed with the \(C^{1}\)-topology. This paper deals with maps which need not be invertible. A point~\(x\in M\) is called singular, if \(D_{x}f:T_{x}M\to T_{f(x)}M\) is not injective. Define \({\mathcal P}(M)\) as the set of all~\(f\in C^{1}(M)\), such that every periodic point of~\(f\) is hyperbolic. Consider a map~\(f\) in the interior of~\({\mathcal P}(M)\) which satisfies that every singular point of~\(f\) contained in the nonwandering set~\(\Omega (f)\) of~\(f\) is periodic and a sink. In this paper it is proved that \(f\) satisfies Axiom~A and the no cycle property. Moreover, \(f\) is \(C^{1}\)~\(\Omega\)-stable, if and only if \(f\) satisfies strong Axiom~A and the no cycle property. These results extend results by \textit{R. Mañé} [Publ. Math., Inst. Hautes Étud. Sci. 66, 161-210 (1988; Zbl 0678.58022)], \textit{J.~Palis} [Publ. Math., Inst. Hautes Étud. Sci. 66, 211-215 (1988; Zbl 0648.58019)] and \textit{F. Przytycki} [Stud. Math. 60, 61-77 (1977; Zbl 0343.58008)].