DOI QR코드

DOI QR Code

MINIMAL SETS OF PERIODS FOR MAPS ON THE KLEIN BOTTLE

  • Kim, Ju-Young ;
  • Kim, Sung-Sook ;
  • Zhao, Xuezhi
  • Published : 2008.05.31

Abstract

The main results concern with the self maps on the Klein bottle. We obtain the Reidemeister numbers and the Nielsen numbers for all self maps on the Klein bottle. In terms of the Nielsen numbers of their iterates, we totally determine the minimal sets of periods for all homotopy classes of self maps on the Klein bottle.

Keywords

periodic point;minimum set of periods;Nielsen number;Klein bottle

References

  1. L. Alseda, S. Baldwin, J. Llibre, R. Swanson, and W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math. 169 (1995), no. 1, 1-32 https://doi.org/10.2140/pjm.1995.169.1
  2. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, 14. American Mathematical Society, Providence, R.I., 1983
  3. B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems 4 (1998), no. 2, 301-320 https://doi.org/10.3934/dcds.1998.4.301
  4. J. Jezierski and W. Marzantowicz, Homotopy minimal periods for nilmanifold maps, Math. Z. 239 (2002), no. 2, 381-414 https://doi.org/10.1007/s002090100303
  5. S. Kwasik and K. B. Lee, The Nielsen numbers of homotopically periodic maps of infranilmanifolds, J. London Math. Soc. (2) 38 (1988), no. 3, 544-554
  6. S. W. Kim, J. B. Lee, and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J. 178 (2005), 37-53 https://doi.org/10.1017/S0027763000009107
  7. K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math. 168 (1995), no. 1, 157-166 https://doi.org/10.2140/pjm.1995.168.157
  8. J. Llibre, A note on the set of periods for Klein bottle maps, Pacific J. Math. 157 (1993), no. 1, 87-93 https://doi.org/10.2140/pjm.1993.157.87
  9. B. Halpern, Periodic points on Klein bottle, unpublished

Cited by

  1. Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces vol.156, pp.15, 2009, https://doi.org/10.1016/j.topol.2009.07.012
  2. Self-maps on flat manifolds with infinitely many periods vol.32, pp.6, 2012, https://doi.org/10.3934/dcds.2012.32.2223