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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Differentiability of the conjugacy in the Hartman-Grobman Theorem
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by Wenmeng Zhang, Kening Lu and Weinian Zhang PDF
Trans. Amer. Math. Soc. 369 (2017), 4995-5030 Request permission

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
  • MR Author ID: 892016
  • Email: mathzwm@sina.com
  • Kening Lu
  • Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
  • MR Author ID: 232817
  • Email: klu@math.byu.edu
  • Weinian Zhang
  • Affiliation: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
  • MR Author ID: 259735
  • Email: matzwn@163.com
  • 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 4995-5030
  • MSC (2010): Primary 35B40; Secondary 35B41, 37L30
  • DOI: https://doi.org/10.1090/tran/6810
  • MathSciNet review: 3632558