Unstable manifolds and Hölder structures associated with noninvertible maps
type: Item
instance of: scholarly article
title: Unstable manifolds and Hölder structures associated with noninvertible maps
zbMATH Open document ID: 1122.37026
author: Eugen Mihailescu
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-02-15T00:00:00Z
review text: The author tries to generalize the theory of dynamical systems from diffeomorphisms to non-invertible smooth maps. Let \(f: M\to M\) be a smooth map of a compact Riemannian manifold \(M\) into itself. The nonwandering set \(\Omega= \Omega_f\) of \(f\) is defined as usual. Assume that \(f\) has Axiom A, i.e. the periodic points of \(f\) are dense in \(\Omega\) and we have a splitting of the tangent bundle over the \(\widehat\Omega\), defined below, into subbundles, one of which is \(E^s\), consisting of contracting directions for the derivative \(Df\) and the other \(E^u\), consisting of the expanding directions for \(Df\). For a compact metric space \((X,d)\) and a continuous map \(f: X\to X\) we define the natural extension \(\widehat X\) of \(X\) with respect to \(f\) as \(\widehat X= \{\widehat x\mid\widehat x= (x,x_{-1}, x_{-2},\dots)\), \(f(x_{-i})= x_{i+1}\,(i\geq 1)\}\). The shift map on \(\widehat X\) is \(\widehat f:\widehat X\to\widehat X\), defined by \(\widehat f(\widehat x)= (f(x), x,x_{-1},\dots)\) and the canonical projection map \(\pi:\widehat X\to X\) is given by \(\pi(\widehat x)= x\) for \(\widehat x= (x,x_{-1},\dots)\in\widehat X\). There exists a natural metric and \(\widehat X\), for every \(K> 1\), given by \(d_K(\widehat x,\widehat y)= d(x,y)+ d(x_{-1}, y_{-1})/K+ d(x_{-2}, y_{-2})/K^2+\cdots\) for \(\widehat x,\widehat y\in\widehat X\). A continuous map \(f: X\to X\) is said to be (topologically) transitive on \(X\) if for any nonempty open sets \(U\), \(V\) in \(X\) there exists \(n\in\mathbb{Z}\) such that \(f^n(U)\cap V\neq\emptyset\). If \(f: M\to M\) is an Axiom A map as above, by the spectral decomposition theorem \(\Omega_f\) can be partitoned into a finite number of \(f\)-invariant subsets, on which \(f\) is transitive. These sets are unique up to order and are called basic sets. We say that a basic set \(\Lambda\) is of saddle type if there are both stable and unstable directions on \(\Lambda\). Now, given a map \(f: M\to M\) with A…
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37D10
zbMATH DE Number: 5127230
zbMATH Keywords: Axiom A maps
zbMATH Keywords: unstable manifolds
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
full work available at URL: dcds.2006.14.419
OpenAlex ID: W1982977132
DOI: 10.3934/DCDS.2006.14.419
description: scientific article; zbMATH DE number 5127230