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June  2012, 32(6): 2223-2232. doi: 10.3934/dcds.2012.32.2223

Self-maps on flat manifolds with infinitely many periods

Zhibin Liang 1, and  Xuezhi Zhao 2,

1. 

Department of Mathematics, Capital Normal University, Beijing 100048, Beijing International Center for Mathematical Research, China
2. 

Department of Mathematics & Institute of mathematics and interdisciplinary science, Capital Normal University, Beijing 100048, China

Received  April 2011 Revised  October 2011 Published  February 2012

This paper deals with the homotopical minimal period of self-maps. We obtain some conditions for self-maps on flat manifolds to guarantee that their homotopical minimal periods are infinite sets.
Keywords: period point., self-map, Homotopical minimal period, flat manifold.
Mathematics Subject Classification: Primary: 37C25; Secondary: 55M2.
Citation: Zhibin Liang, Xuezhi Zhao. Self-maps on flat manifolds with infinitely many periods. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2223-2232. doi: 10.3934/dcds.2012.32.2223
References:
[1]

L. Alsedà, S. Baldwin, J. Llibre, R. Swanson and W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math., 169 (1995), 1-32.  Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems" (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, (1980), 18-34.  Google Scholar

[3]

L. Charlap, "Bieberbach Groups and Flat Manifolds," Universitext, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2.  Google Scholar

[4]

J. Jezierski, Wecken's theorem for periodic points in dimension at least $3$, Topology and its Applications, 153 (2006), 1825-1837. doi: 10.1016/j.topol.2005.06.008.  Google Scholar

[5]

J. Jezierski, E. Keppelmann and W. Marzantowicz, Wecken property for periodic points on the Klein bottle, Topol. Methods Nonlinear Anal., 33 (2009), 51-64.  Google Scholar

[6]

B. Jiang, "Lectures on Nielsen Fixed Point Theory," Contemporary Mathematics, 14, American Mathematical Society, Providence, R.I., 1983.  Google Scholar

[7]

B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems, 4 (1998), 301-320. doi: 10.3934/dcds.1998.4.301.  Google Scholar

[8]

J. Jezierski and W. Marzantowicz, "Homotopy Methods in Topological Fixed and Periodic Points Theory," Topological Fixed Point Theory and Its Applications, 3, Springer, Dordrecht, 2006.  Google Scholar

[9]

J. Y. Kim, S. S. Kim and X. Zhao, Minimal sets of periods for maps on the Klein bottle, J. Korean Math. Soc., 45 (2008), 883-902. doi: 10.4134/JKMS.2008.45.3.883.  Google Scholar

[10]

S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J., 178 (2005), 37-53.  Google Scholar

[11]

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166.  Google Scholar

[12]

J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011-2023. doi: 10.1016/j.geomphys.2005.11.003.  Google Scholar

[13]

J. B. Lee and X. Zhao, Homotopy minimal periods for expanding maps on infra-nilmanifolds, J. Math. Soc. Japan, 59 (2007), 179-184. doi: 10.2969/jmsj/1180135506.  Google Scholar

[14]

J. Llibre, A note on the set of periods for Klein bottle maps, Pacific J. Math., 157 (1993), 87-93.  Google Scholar

[15]

R. Tauraso, Sets of periods for expanding maps on flat manifolds, Monatshefte für Mathematik, 128 (1999), 151-157. doi: 10.1007/s006050050052.  Google Scholar

show all references

References:
[1]

L. Alsedà, S. Baldwin, J. Llibre, R. Swanson and W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math., 169 (1995), 1-32.  Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems" (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, (1980), 18-34.  Google Scholar

[3]

L. Charlap, "Bieberbach Groups and Flat Manifolds," Universitext, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2.  Google Scholar

[4]

J. Jezierski, Wecken's theorem for periodic points in dimension at least $3$, Topology and its Applications, 153 (2006), 1825-1837. doi: 10.1016/j.topol.2005.06.008.  Google Scholar

[5]

J. Jezierski, E. Keppelmann and W. Marzantowicz, Wecken property for periodic points on the Klein bottle, Topol. Methods Nonlinear Anal., 33 (2009), 51-64.  Google Scholar

[6]

B. Jiang, "Lectures on Nielsen Fixed Point Theory," Contemporary Mathematics, 14, American Mathematical Society, Providence, R.I., 1983.  Google Scholar

[7]

B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems, 4 (1998), 301-320. doi: 10.3934/dcds.1998.4.301.  Google Scholar

[8]

J. Jezierski and W. Marzantowicz, "Homotopy Methods in Topological Fixed and Periodic Points Theory," Topological Fixed Point Theory and Its Applications, 3, Springer, Dordrecht, 2006.  Google Scholar

[9]

J. Y. Kim, S. S. Kim and X. Zhao, Minimal sets of periods for maps on the Klein bottle, J. Korean Math. Soc., 45 (2008), 883-902. doi: 10.4134/JKMS.2008.45.3.883.  Google Scholar

[10]

S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J., 178 (2005), 37-53.  Google Scholar

[11]

K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166.  Google Scholar

[12]

J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011-2023. doi: 10.1016/j.geomphys.2005.11.003.  Google Scholar

[13]

J. B. Lee and X. Zhao, Homotopy minimal periods for expanding maps on infra-nilmanifolds, J. Math. Soc. Japan, 59 (2007), 179-184. doi: 10.2969/jmsj/1180135506.  Google Scholar

[14]

J. Llibre, A note on the set of periods for Klein bottle maps, Pacific J. Math., 157 (1993), 87-93.  Google Scholar

[15]

R. Tauraso, Sets of periods for expanding maps on flat manifolds, Monatshefte für Mathematik, 128 (1999), 151-157. doi: 10.1007/s006050050052.  Google Scholar

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