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Transactions of the American Mathematical Society

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Differentiability of the conjugacy in the Hartman-Grobman Theorem


Authors: Wenmeng Zhang, Kening Lu and Weinian Zhang
Journal: Trans. Amer. Math. Soc. 369 (2017), 4995-5030
MSC (2010): Primary 35B40; Secondary 35B41, 37L30
DOI: https://doi.org/10.1090/tran/6810
Published electronically: February 13, 2017
MathSciNet review: 3632558
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Abstract: The classical Hartman-Grobman Theorem states that a smooth diffeomorphism $ F(x)$ near its hyperbolic fixed point $ \bar x$ is topological conjugate to its linear part $ DF(\bar x)$ by a local homeomorphism $ \Phi (x)$. In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth $ F(x)$ is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a $ C^\infty $ diffeomorphism $ F(x)$, the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a $ C^1$ diffeomorphism $ F(x)$ with $ DF(x)$ being $ \alpha $-Hölder continuous at the fixed point that the local homeomorphism $ \Phi (x)$ is differentiable at the fixed point. Here, $ \alpha >0$ depends on the bands of the spectrum of $ F'(\bar x)$ for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on $ F(x)$ cannot be lowered to $ C^1$.


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Additional Information

Wenmeng Zhang
Affiliation: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of China
Email: mathzwm@sina.com

Kening Lu
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: klu@math.byu.edu

Weinian Zhang
Affiliation: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
Email: matzwn@163.com

DOI: https://doi.org/10.1090/tran/6810
Keywords: Differentiable linearization, invariant manifold, invariant foliation, bump function.
Received by editor(s): September 10, 2014
Received by editor(s) in revised form: August 20, 2015
Published electronically: February 13, 2017
Additional Notes: This work was partially supported by grants from NSFC grants #11301572, #11231001 and #11221101, an NSF grant, and Chongqing Normal University project #13XLZ04.
Article copyright: © Copyright 2017 American Mathematical Society

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