On smooth conjugacy of expanding maps in higher dimensions

Yong Fang

doi:10.3934/dcds.2011.30.687

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2011, Volume 30, Issue 3: 687-697. Doi: 10.3934/dcds.2011.30.687

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On smooth conjugacy of expanding maps in higher dimensions

Yong Fang^1,

1.
Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex

Received: May 2010

Revised: November 2010

Published: September 2011

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Abstract

In this paper we investigate smooth conjugacy of $C^\infty$ expanding maps on certain nilmanifolds. We show that several rigidity results about expanding maps on the circle can not be generalized directly to higher dimensions. For example the following result is obtained: Let $\Gamma_1\backslash N_1$ and $\Gamma_2\backslash N_2$ be two nilmanifolds of homogeneous type. We show that for any positive integer $k$ there exist on the product nilmanifold $\Gamma_1\times \Gamma_2\backslash N_1\times N_2$ a $C^\infty$ expanding map $\varphi$ and an expanding nilendomorphism $\psi$ which are $C^k$ conjugate, but not $C^{k,lip}$ conjugate. While in the case of dimension one, it was shown by M. Shub and D. Sullivan that if two $C^\infty$ expanding maps on $\mathbb S^1$ are absolutely continuously conjugate, then they must be $C^\infty$ conjugate.

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Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D40.

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References

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