Stable manifolds for nonuniform polynomial dichotomies
type: Item
instance of: scholarly article
title: Stable manifolds for nonuniform polynomial dichotomies
zbMATH Open document ID: 1194.47017
author: António J. G. Bento
author: César M. Silva
published in: Journal of Functional Analysis
publication date: 2009-06-30T00:00:00Z
review text: Let \(X\) be a Banach space and \((A_n)_n\) a sequence of invertible bounded linear operators on \(X\). For \(m\geq n\), define \[ A(m,n) := \begin{cases} A_{m-1}\cdots A_n & \text{if \(m>n\)}, \\ \text{Id} & \text{if \(m=n\).} \end{cases} \] In this article, the authors introduce the notion of nonuniform polynomial dichotomy and ask for the existence of smooth manifolds in \(X\) which are stable under the dynamics of a perturbation of \(\{ A(m,n) \}\) if \((A_n)\) admits a nonuniform polynomial dichotomy. More precisely, the sequence \((A_n)\) is said to admit a nonuniform polynomial dichotomy if there exists a sequence \((P_n)\) of projections on \(X\) which commute with \(\{ A(m,n) \}\), i.e., \(P_m A(m,n) = A(m,n) P_n\), and there exist \(\epsilon\geq 0\), \(b\geq 0 > a\), \(D\geq 1\) such that for \(m\geq n\), we have \[ \| A(m,n) P_n \| \leq D(m-n+1)^a n^\epsilon \] and \[ \| A(m,n)^{-1} (\text{Id} - P_m) \| \leq D (m-n+1)^{-b} m^\epsilon. \] For a sequence of perturbations \((f,n)\), \(f_n: X\to X\), the authors consider the perturbed family \[ F(m,n) := \begin{cases} (A_{m-1} + f_{m-1})\cdots (A_n + f_n) & \text{if \(m>n\)}, \\ \text{Id} & \text{if \(m=n\).} \end{cases} \] For one family of perturbations, defined via growth restrictions of polynomial type, they show the existence of local Lipschitz stable manifolds. For a more restrictive family of perturbations, they establish the existence of global \(C^1\) stable manifolds. For both families, they provide explicit examples.
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37D25
zbMATH DE Number: 5572189
zbMATH Keywords: invariant manifolds
zbMATH Keywords: nonautonomous dynamics
zbMATH Keywords: nonuniform polynomial dichotomies
reviewed by: Anke D. Pohl
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.01.032
OpenAlex ID: W2059547075
cites work: Q5431024
cites work: Integral stable manifolds in Banach spaces
cites work: Regularity of invariant manifolds for nonuniformly hyperbolic dynamics
cites work: Higher regularity of invariant manifolds for nonautonomous equations
cites work: Smoothness of invariant manifolds for nonautonomous equations
cites work: Existence of stable manifolds for nonuniformly hyperbolic \(C^1\) dynamics
cites work: Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy
cites work: Stable manifolds for nonautonomous equations without exponential dichotomy
cites work: Stability theory and Lyapunov regularity
cites work: Characterization of stable manifolds for nonuniform exponential dichotomies
cites work: Growth rates and nonuniform hyperbolicity
cites work: Ck centre unstable manifolds
cites work: On sequences of 𝐶^{𝑘,𝛿}_{𝑏} maps which converge in the uniform 𝐶⁰-norm
cites work: Geometric theory of semilinear parabolic equations
cites work: Q5179757
cites work: Q3672767
cites work: Q5646734
cites work: FAMILIES OF INVARIANT MANIFOLDS CORRESPONDING TO NONZERO CHARACTERISTIC EXPONENTS
cites work: CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY
cites work: GEODESIC FLOWS ON CLOSED RIEMANNIAN MANIFOLDS WITHOUT FOCAL POINTS
cites work: Ergodic Attractors
cites work: Ergodic theory of differentiable dynamical systems
cites work: Characteristic exponents and invariant manifolds in Hilbert space
cites work: Dynamics of evolutionary equations
cites work: Fibres dynamiques asymptotiquement compacts, exposants de Lyapunov. Entropie. Dimension. (Asymptotically compact dynamical bundles, Lyapunov exponents. Entropy. Dimension)
DOI: 10.1016/J.JFA.2009.01.032
description: scientific article; zbMATH DE number 5572189