Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On smooth conjugacy of expanding maps in higher dimensions

Pages: 687 - 697, Volume 30, Issue 3, July 2011      doi:10.3934/dcds.2011.30.687

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Yong Fang - Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France (email)

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.

Keywords:  Expanding map, smooth conjugacy, entropy.
Mathematics Subject Classification:  Primary: 37A35, 34D20; Secondary: 37D35, 37D40.

Received: May 2010;      Revised: November 2010;      Available Online: March 2011.