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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18. Google Scholar

[3]

L. Charlap, "Bieberbach Groups and Flat Manifolds,", Universitext, (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. 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. Google Scholar

[6]

B. Jiang, "Lectures on Nielsen Fixed Point Theory,", Contemporary Mathematics, 14 (1983). Google Scholar

[7]

B. Jiang and J. Llibre, Minimal sets of periods for torus maps,, Discrete Contin. Dynam. Systems, 4 (1998), 301. 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 (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. 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. Google Scholar

[11]

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

[15]

R. Tauraso, Sets of periods for expanding maps on flat manifolds,, Monatshefte für Mathematik, 128 (1999), 151. 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. Google Scholar

[2]

L. Block, J. Guckenheimer, M. Misiurewicz and L. Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18. Google Scholar

[3]

L. Charlap, "Bieberbach Groups and Flat Manifolds,", Universitext, (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. 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. Google Scholar

[6]

B. Jiang, "Lectures on Nielsen Fixed Point Theory,", Contemporary Mathematics, 14 (1983). Google Scholar

[7]

B. Jiang and J. Llibre, Minimal sets of periods for torus maps,, Discrete Contin. Dynam. Systems, 4 (1998), 301. 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 (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. 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. Google Scholar

[11]

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

[15]

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

[1]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

[2]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507

[3]

Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961

[4]

Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337

[5]

Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure & Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175

[6]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555

[7]

Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227

[8]

Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029

[9]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933

[10]

Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41

[11]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795

[12]

Linping Peng, Yazhi Lei. The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 873-890. doi: 10.3934/dcds.2011.30.873

[13]

Guibin Lu, Qiying Hu, Youying Zhou, Wuyi Yue. Optimal execution strategy with an endogenously determined sales period. Journal of Industrial & Management Optimization, 2005, 1 (3) : 289-304. doi: 10.3934/jimo.2005.1.289

[14]

Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1269-1283. doi: 10.3934/cpaa.2012.11.1269

[15]

Marcelo Marchesin. The mass dependence of the period of the periodic solutions of the Sitnikov problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 597-609. doi: 10.3934/dcdss.2008.1.597

[16]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073

[17]

Reza Lotfi, Gerhard-Wilhelm Weber, S. Mehdi Sajadifar, Nooshin Mardani. Interdependent demand in the two-period newsvendor problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-24. doi: 10.3934/jimo.2018143

[18]

Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences & Engineering, 2011, 8 (4) : 931-952. doi: 10.3934/mbe.2011.8.931

[19]

Brandy Rapatski, Petra Klepac, Stephen Dueck, Maoxing Liu, Leda Ivic Weiss. Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000. Mathematical Biosciences & Engineering, 2006, 3 (3) : 545-556. doi: 10.3934/mbe.2006.3.545

[20]

Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial & Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]