Differentiability of the conjugacy in the Hartman-Grobman Theorem
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- by Wenmeng Zhang, Kening Lu and Weinian Zhang PDF
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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 $C^1$.References
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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