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

Self-maps on flat manifolds with infinitely many periods

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.
Citation: Zhibin Liang, Xuezhi Zhao. Self-maps on flat manifolds with infinitely many periods. Discrete and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[15]

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

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[15]

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

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