Browse > Article
http://dx.doi.org/10.4134/CKMS.c170304

HOMOTOPY MINIMAL PERIODS OF MAPS ON THE KLEIN BOTTLE  

Lee, Jong Bum (Department of Mathematics Sogang University)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.3, 2018 , pp. 961-971 More about this Journal
Abstract
We determine the sets of homotopy minimal periods of all self-maps on the Klein bottle by using a single formula for homotopy minimal periods of maps on the infra-solvmanifolds of type (R). This provides an alternate but an easy proof for the main results of [12].
Keywords
homotopy minimal period; infra-solvmanifold; Klein bottle;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 L. Block, J. Guckenheimer, M. Misiurewicz, and L. S. Young, Periodic points and topological entropy of one-dimensional maps, in Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), 18-34, Lecture Notes in Math., 819, Springer, Berlin, 1980.
2 A. Fel'shtyn and J. B. Lee, The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type (R), Topology Appl. 181 (2015), 62-103.   DOI
3 J. W. Hoffman, Z. Liang, Y. Sakai, and X. Zhao, Homotopy minimal period self-maps on flat manifolds, Adv. Math. 248 (2013), 324-334.   DOI
4 G. J. O. Jameson, The cyclotomic polynomials, http://www.maths.lancs.ac.uk/jameson.
5 J. Jezierski, J. Kedra, and W. Marzantowicz, Homotopy minimal periods for NR-solvmanifolds maps, Topology Appl. 144 (2004), no. 1-3, 29-49.   DOI
6 J. Jezierski, E. Keppelmann, and W. Marzantowicz, Wecken property for periodic points on the Klein bottle, Topol. Methods Nonlinear Anal. 33 (2009), no. 1, 51-64.   DOI
7 J. Jezierski, Homotopy minimal periods for maps of three-dimensional nilmanifolds, Pacific J. Math. 209 (2003), no. 1, 85-101.   DOI
8 B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems 4 (1998), no. 2, 301-320.   DOI
9 J. H. Jo and J. B. Lee, Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with Sol41-geometry, Fixed Point Theory Appl. 2015 (2015), 175, 15 pp.   DOI
10 H. J. Kim, J. B. Lee, and W. S. Yoo, Computation of the Nielsen type numbers for maps on the Klein bottle, J. Korean Math. Soc. 45 (2008), no. 5, 1483-1503.   DOI
11 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), no. 3, 883-902.   DOI
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), no. 10, 2011-2023.   DOI
13 J. B. Lee and X. Zhao, Homotopy minimal periods for expanding maps on infra-nilmanifolds, J. Math. Soc. Japan 59 (2007), no. 1, 179-184.   DOI
14 J. B. Lee, Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds, Sci. China Ser. A 51 (2008), no. 3, 351-360.   DOI
15 J. B. Lee, Nielsen type numbers and homotopy minimal periods for maps on 3-solvmanifolds, Algebr. Geom. Topol. 8 (2008), no. 1, 563-580.   DOI
16 J. B. Lee, Density of the homotopy minimal periods of maps on infra-solvmanifolds of type (R), J. Korean Math. Soc. 55 (2018), no. 2, 293-311.   DOI
17 J. Jezierski and W. Marzantowicz, Homotopy minimal periods for nilmanifold maps, Math. Z. 239 (2002), no. 2, 381-414.   DOI
18 J. Llibre, A note on the set of periods for Klein bottle maps, Pacific J. Math. 157 (1993), no. 1, 87-93.   DOI
19 K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math. 168 (1995), no. 1, 157-166.   DOI
20 Z. Liang and X. Zhao, Self-maps on flat manifolds with infinitely many periods, Discrete Contin. Dyn. Syst. 32 (2012), no. 6, 2223-2232.   DOI
21 O. M. Sarkovs'kii, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Z. 16 (1964), 61-71.