\(\sigma\)-Hölder continuous linearization near hyperbolic fixed points in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: \(\sigma\)-Hölder continuous linearization near hyperbolic fixed points in \(\mathbb{R}^n\)
zbMATH Open document ID: 0978.37012
author: Bin Tan
published in: Journal of Differential Equations
publication date: 2001-10-24T00:00:00Z
review text: The paper is devoted to the study of problems related to regularity of a local linearization for a smooth nonlinear map \(T:\mathbb R^n\to\mathbb R^n\) near its hyperbolic fixed point. Recall that the classical Grobman-Hartman theorem guarantees only the existence of a continuous linearization \(H\) of the map \(T\) near the hyperbolic fixed point \(x=0\). On the other hand, the nonexistence results for \(C^1\)-linearization are also well known [see e.g. \textit{P. Hartman}, Ordinary differential equations, Wiley, New York (1964; Zbl 0125.32102). In the present paper the possibility of \(\sigma\)-Hölder continuous linearization of any \(C^{1,1}\)-map \(T\) near its hyperbolic fixed point \(x=0\), where the exponent \(\sigma\in(0,1)\) can be explicitly expressed in terms of the spectrum of \(D_xT(0)\), is proved. Using this explicit expression the author formulates conditions, which guarantee the existence of \(\sigma\)-Hölder continuous linearization with any \(\sigma\in(0,1)\). In particular, it is shown that for \(3\)-dimensional maps \(T:\mathbb R^3\to\mathbb R^3\) the \(\sigma\)-Hölder continuous linearization with any \(\sigma\in (0,1)\) exists without additional restrictions on \(T\).
reviewed by: S. V. Zelik
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 1454734
zbMATH Keywords: Hölder continuous linearization
zbMATH Keywords: Grobman-Hartman theorem
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3693
OpenAlex ID: W2068961441
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DOI: 10.1006/JDEQ.1999.3693
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description: scientific article; zbMATH DE number 1454734