Strange attractors in saddle-node cycles: Prevalence and globality
type: Item
instance of: scholarly article
title: Strange attractors in saddle-node cycles: Prevalence and globality
zbMATH Open document ID: 0865.58034
author: Marcelo Viana
author: L. J. Díaz
author: Jorge Rocha
published in: Inventiones Mathematicae
publication date: 1997-07-13T00:00:00Z
review text: The paper deals with parametrized families of diffeomorphisms bifurcating through the creation of critical saddle-node cycles. The authors prove that: 1. If \(\{f_\mu\}_\mu\) is a generic smooth family of diffeomorphisms on a manifold \(M\), unfolding a critical saddle-node cycle and \(A\) is the set of all values of \(\mu\) for which \(f_\mu\) exhibits Hénon-like strange attractors then \[ \liminf_{\varepsilon\to 0} \frac{m(A\cap (-\varepsilon,\varepsilon))}{2\varepsilon}>0. \] 2. Let \(B\) be the set of values of \(\mu\) for which \[ \overline{\bigcup_{x\in M}\alpha(x)}\cup \overline{\bigcup_{x\in M} \omega(x)} \] is a hyperbolic set of \(f_\mu\). Then, there exists an open set of smooth families of diffeomorphisms \(\{f_\mu\}_{\mu\in\mathbb{R}}\) unfolding a critical saddle-node cycle, for which \(B\) satisfies \[ \liminf_{\varepsilon\to\infty} \frac{m(B\cap (-\varepsilon, \varepsilon))}{2\varepsilon}>0. \]
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 885520
zbMATH Keywords: hyperbolicity of cycles
zbMATH Keywords: saddle-node cycles
zbMATH Keywords: Hénon-like strange attractors
reviewed by: S. I. Nenov
MaRDI profile type: Publication
full work available at URL: s002220050068
OpenAlex ID: W2013602841
DOI: 10.1007/S002220050068
description: scientific article; zbMATH DE number 885520