Minimal non-hyperbolicity and index-completeness
type: Item
instance of: scholarly article
title: Minimal non-hyperbolicity and index-completeness
zbMATH Open document ID: 1214.37021
author: Shaobo Gan
author: Lan Wen
author: Da-Wei Yang
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-12-16T00:00:00Z
review text: Let \(M\) be a compact \(d\)-dimensional \(C^\infty\)-manifold without boundary and \(\text{Diff}^1(M)\) the space of \(C^1\)-diffeomorphisms from \(M\) into itself endowed with the \(C^1\)-topology. Let \(p\in M\) be a hyperbolic periodic point and denote by \(H(p,f)\) (the homoclinic class of \(p\)) defined as the closure of all transverse intersections of the stable and unstable manifolds of the orbit of \(p\). Also the index of \(p\), denoted by \(\text{ind}(p)\) is the dimension of its stable manifold. For a \(C^1\) generic \(f\) (which means, for an \(f\) belonging to a \(C^1\)-residual set of \(\text{Diff}^1(M)\)) it was proved by \textit{F. Abdenur, Ch. Bonatti, S. Crovisier, L J. Díaz} and \textit{L. Wen} [Ergodic Theory Dyn. Syst. 27, No. 1, 1--22 (2007; Zbl 1144.37007)] that every homoclinic class \(H(p,f)\) containing hyperbolic periodic points of indices \(\alpha\) and \(\beta\) contains also a dense set of hyperbolic periodic points of index \(\gamma\) for all \(\alpha< \gamma< \beta\). Such result means that for a \(C^1\)-generic \(f\), every homoclinic class is index-complete in the sense that the indexes of periodic points contained in \(H(p,f)\) form an interval in \(\mathbb N\). Since whether \(H(f,p)\) is an hyperbolic set, it would contain points with a unique index, therefore the former result is referred to a non-hyperbolic setting. A preliminary result was given by \textit{C. Bonatti, L. J. Díaz, E. R. Pujals} and \textit{J. Rocha} [Astérisque 286, 187--222 (2003; Zbl 1056.37024)], in the sense that if \(f\) is a \(C^1\) generic diffeomorphism, then every robust transitive set (that is, an isolated invariant set remaining transitive by small perturbations) is index-complete. Another related problem is to consider the set of indexes of hyperbolic periodic orbits that approach to, but not necessarily contained in a given chain transitive set \(\Lambda\). One chain transitive set is a compact and invariant set contained in \(M\) for which for ev…
reviewed by: Francisco Balibrea
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 5648564
zbMATH Keywords: dimension theory
zbMATH Keywords: Poincaré recurrences
zbMATH Keywords: multifractal analysis
MaRDI profile type: Publication
full work available at URL: dcds.2009.25.1349
OpenAlex ID: W1998944263
DOI: 10.3934/DCDS.2009.25.1349
description: scientific article; zbMATH DE number 5648564