The creation of homoclinic points of \(C^ 1\)-maps (Q1314666)

Let \(M\) be a closed \(C^ \infty\)-manifold, \(f: M \mapsto M\) a \(C^ 1\)- map and \(p \in M\) a hyperbolic fixed point of \(f\). The points of intersection of the stable and the unstable manifolds of \(p\) are called homoclinic points. The points of intersection of the stable manifold and the closure of the unstable manifold and those of the unstable manifold and the closure of the stable manifold (different from \(p\)) are called almost hyperbolic points. Extending a result due to \textit{R. Ma\=né} [Publ. Math., Inst. Hautes Étud. Sci. 66, 139-159 (1988; Zbl 0669.58024)] concerning diffeomorphisms the authors prove that if the map \(f\) has almost homoclinic points then by arbitrary small perturbations in the \(C^ 1\)-topology one can create homoclinic points.