Breaking cycles on surfaces (Q791951)

Newhouse and Palis proposed the question of whether a diffeomorphism on a compact manifold with hyperbolic nonwandering set can be approximated by an \(\Omega\)-stable one. They also proved, in the two-dimensional case, that such a diffeomorphism can be perturbed in such a way as to obtain an \(\Omega\)-stable one which has the same nonwandering set [\textit{S. Newhouse} and \textit{J. Palis}, Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 293-301 (1973; Zbl 0279.58010)]. \textit{M. Kurata} gave counterexamples of diffeomorphisms on manifolds of di\(m\geq 4\) [Nagoya Math. J. 74, 77-86 (1979; Zbl 0388.58017)]. \textit{A. Dankner} gave counterexamples on manifolds of di\(m\geq 3\) [Topology 19, 163-177 (1980; Zbl 0458.58013)]. In this paper the authors give a positive answer, in the two-dimensional case, to the following more general question: can a diffeomorphism with hyperbolic limit set be approximated by an \(\Omega\)- stable one with the same limit set?