On inverse limit stability for maps
type: Item
instance of: scholarly article
title: On inverse limit stability for maps
zbMATH Open document ID: 0677.58028
DOI: 10.1016/0022-0396(89)90106-X
author: Juergen Quandt
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: Two not necessarily invertible maps f and g from a Banach manifold X into itself are said to be inverse limit conjugate if the shifts on the respective spaces of orbits \(\{x\in X^{{\mathbb{Z}}}| \quad f(x_ i)=x_{i+1},\quad i\in {\mathbb{Z}}\}\) and \(\{x\in X^{{\mathbb{Z}}}| \quad g(x_ i)=x_{i+1},\quad i\in {\mathbb{Z}}\}\) are conjugate in the usual sense: there exists a homeomorphism between the spaces of orbits (as subsets of \(X^{{\mathbb{Z}}}\) with the product topology), and this homeomorphism maps orbits to orbits. If f and g are homeomorphisms then inverse limit conjugacy and conjugacy are equivalent. A \(C^ r\) map, \(r\geq 1\), is said to be \(C^ r\) inverse limit stable if it lies in the \(C^ r\) interior of its equivalence class (with respect to inverse limit conjugacy). The following results are presented: A hyperbolic linear map L with skewness \(a=\max \{| L_ s|,| L_ u^{-1}| \}\) is inverse limit conjugate to Lipschitz perturbations of the form \(L+f\), provided the Lipschitz constant of f does not exceed 1-a (Theorem 2.1). Suppose a \(C^ 1\) map f has a hyperbolic invariant set I such that \(\cap_{n\in {\mathbb{N}}}f^ n(I)\) is compact. Then for all g from a \(C^ 1\) neighbourhood of f there exists an invariant set J such that f and g are inverse limit conjugate with respect to the spaces of orbits over I and J, respectively. The conjugating homeomorphism can be made arbitrarily close to the identity by choosing the neighbourhood of f sufficiently small (Theorem 3.1). From Theorem 3.1 it is deduced that ``Anosov maps'' are inverse limit stable and form an open subset of the \(C^ 1\) maps of X. Several conjectures concerning global stability, a spectral decomposition, Axiom A and Morse-Smale maps, and \(\Omega\)- inverse limit stability are formulated. Then the continuous time case (semiflows instead of maps) is discussed; analogous results are presented. Finally, applications to evolution equations are mentioned very briefly.
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 4109690
zbMATH Keywords: inverse limit stability
zbMATH Keywords: orbits
zbMATH Keywords: inverse limit conjugacy
zbMATH Keywords: Lipschitz perturbations
zbMATH Keywords: hyperbolic invariant set
zbMATH Keywords: Axiom A
zbMATH Keywords: Morse-Smale maps
reviewed by: Hans Crauel
MaRDI profile type: Publication
cites work: "Almost" Implies "Near"
cites work: Q5566243
cites work: An introduction to infinite dimensional dynamical systems -- geometric theory. With an appendix by Krzysztof P. Rybakowski
cites work: Q3917197
cites work: Hyperbolic sets for semilinear parabolic equations
cites work: Q4076563
cites work: A weak Hartman's theorem for homomorphisms and semi-groups in a Banach space
cites work: Stability of Morse-Smale maps
cites work: On Morse-Smale dynamical systems
cites work: Anosov endomorphisms
cites work: On Ω-stability and structural stability of endomorphisms satisfying Axiom A
cites work: On a Theorem of P. Hartman
cites work: On structural stability for semiflows
cites work: On the Hartman-Grobman theorem for maps
cites work: Stability of Anosov Maps
cites work: A structural stability theorem
cites work: Structural stability of \(C^1\) diffeomorphisms
cites work: Differentiable dynamical systems
cites work: Q5605743
cites work: Q4039918
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description: scientific article; zbMATH DE number 4109690