How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency
type: Item
instance of: scholarly article
title: How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency
zbMATH Open document ID: 1110.37034
published in: Advances in Mathematics
publication date: 2007-01-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 5083649
zbMATH Keywords: homoclinic tangency
zbMATH Keywords: newhouse domain
zbMATH Keywords: infinitely many coexisting sinks
zbMATH Keywords: prevalence
zbMATH Keywords: non-uniform hyperbolicity
author: Anton Gorodetski
author: Vadim Kaloshin
reviewed by: A. Klíč
MaRDI profile type: Publication
full work available at URL: j.aim.2006.03.012
OpenAlex ID: W1983783500
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DOI: 10.1016/J.AIM.2006.03.012
description: scientific article; zbMATH DE number 5083649