# American Institute of Mathematical Sciences

2011, 30(3): 687-697. doi: 10.3934/dcds.2011.30.687

## On smooth conjugacy of expanding maps in higher dimensions

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

Received  May 2010 Revised  November 2010 Published  March 2011

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.
Citation: Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687
##### References:
 [1] K. Dekimpe and K. B. Lee, Expanding maps on infra-nilmanifolds of homogeneous type,, Trans. Amer. Math. Soc., 355 (2003), 1067. doi: 10.1090/S0002-9947-02-03084-2. [2] R. Feres, Hyperbolic dynamical systems, invariant geometric structures, and rigidity,, Math. Res. Lett., 1 (1994), 11. [3] R. Feres, The invariant connection of a 1/2-pinched Anosov diffeomorphism and rigidity,, Pacific J. Math., 171 (1995), 139. [4] M. Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Etudes Sci. Publ. Math., 53 (1981), 53. doi: 10.1007/BF02698687. [5] P. Hall, "Nilpotent Groups,", Queen Mary College Maths. Notes, (1969). [6] K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups,, Contemporary Math. A. M. S., 44 (1985), 73. [7] R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,, Commun. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. [8] M. Misiurewicz, On expanding maps of compact manifolds and local homeomorphisms of a circle,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 18 (1970), 725. [9] R. Sacksteder, The measures invariant under an expanding map,, Géométrie différentielle, 392 (1974), 179. [10] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. doi: 10.2307/2373276. [11] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited,, Ergodic Theory Dynam. Systems, 5 (1985), 285. doi: 10.1017/S014338570000290X.

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##### References:
 [1] K. Dekimpe and K. B. Lee, Expanding maps on infra-nilmanifolds of homogeneous type,, Trans. Amer. Math. Soc., 355 (2003), 1067. doi: 10.1090/S0002-9947-02-03084-2. [2] R. Feres, Hyperbolic dynamical systems, invariant geometric structures, and rigidity,, Math. Res. Lett., 1 (1994), 11. [3] R. Feres, The invariant connection of a 1/2-pinched Anosov diffeomorphism and rigidity,, Pacific J. Math., 171 (1995), 139. [4] M. Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Etudes Sci. Publ. Math., 53 (1981), 53. doi: 10.1007/BF02698687. [5] P. Hall, "Nilpotent Groups,", Queen Mary College Maths. Notes, (1969). [6] K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups,, Contemporary Math. A. M. S., 44 (1985), 73. [7] R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,, Commun. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. [8] M. Misiurewicz, On expanding maps of compact manifolds and local homeomorphisms of a circle,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 18 (1970), 725. [9] R. Sacksteder, The measures invariant under an expanding map,, Géométrie différentielle, 392 (1974), 179. [10] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. doi: 10.2307/2373276. [11] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited,, Ergodic Theory Dynam. Systems, 5 (1985), 285. doi: 10.1017/S014338570000290X.
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