How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency
This enlarged and interesting paper deals with the question: how often does a surface diffeomorphism have infinite many sinks. Let \(M\) be a smooth 2-dimensional manifold and \(\text{Diff}^r(M)\) be the space of \(C^r\)-smooth diffeomorphisms. A diffeomorphism \(f\in\text{Diff}^r(M)\) exhibits a homoclinic tangency (HT) if it has a saddle periodic point \(p= f^n(p)\), such that \(W^s(p)\) and \(W^u(p)\) have a point \(q\) of the tangency. It is known that an open set \(U\subset\text{Diff}^r(M)\) (Newhouse domain) exists such that there is a Baire set of diffeomorphisms having infinitely many coexisting sinks. Here, the periodic trajectories of special kind (\(V\)-localized ) are defined and as well as the notion of the cyclicity of a \(V\)-localized periodic orbit. One of main results here is: For a generic 1-parameter family \(\{f_\varepsilon\}\) that unfolds an HT at \(q\), there is a sequence of numbers \(\{N_s: s\in\mathbb{N}\}\) such that for almost every parameter \(\varepsilon\) and any \(D\in\mathbb{N}\), the corresponding \(f_\varepsilon\) has only finitely many \(V\)-localized sinks \(\{P_j\}\) whose cyclicity is bounded by \(D\) or period exceeds \(\mathbb{N}_{s_j}\), where \(s_j> D\) is cyclicity of a corresponding sink \(P_j\). In other words, for almost every parameter \(\varepsilon\), if there are infinitely many coexisting \(V\)-localized sinks \(\{P_j\}\), then all but finitely many have cyclicity \(s_j> D\) and period \(<\mathbb{N}_{s_j}\). The paper is well written and organized.
scientific article; zbMATH DE number 5083649
scientific article