Browse > Article
http://dx.doi.org/10.4134/JKMS.j170189

DENSITY OF THE HOMOTOPY MINIMAL PERIODS OF MAPS ON INFRA-SOLVMANIFOLDS OF TYPE (R)  

Lee, Jong Bum (Department of Mathematics Sogang University)
Zhao, Xuezhi (Department of Mathematics Institute of Mathematics and Interdisciplinary Science Capital Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.2, 2018 , pp. 293-311 More about this Journal
Abstract
We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). We give some examples of maps on infra-solvmanifolds of dimension three for which the corresponding density is positive.
Keywords
holonomy; homotopy minimal period; infra-solvmanifold; periodic point;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 J. B. Lee and X. Zhao, Nielsen type numbers and homotopy minimal periods for maps on 3-solv-manifolds, Algebr. Geom. Topol. 8 (2008), no. 1, 563-580.   DOI
2 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
3 J. Llibre, A note on the set of periods for Klein bottle maps, Pacific J. Math. 157 (1993), no. 1, 87-93.   DOI
4 D. S. Passman, The Algebraic Structure of Group Rings, Pure and Appl. Math., Wiley, New York, 1977.
5 O. M. Sarkovs'kii, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Z. 16 (1964), 61-71.
6 R. Tauraso, Sets of periods for expanding maps on flat manifolds, Monatsh. Math. 128 (1999), no. 2, 151-157.   DOI
7 B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2000), no. 2, 195-237.   DOI
8 J. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Wilmington, 1984.
9 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.   DOI
10 L. Block, J. Guckenheimer, M. Misiurewicz, and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 18-34, Lecture Notes in Math., 819, Springer, Berlin, 1980.
11 D. Choi and J. Shin, Free actions of finite Abelian groups on 3-dimensional nilmanifolds, J. Korean Math. Soc. 42 (2005), no. 4, 795-826.   DOI
12 A. Fel'shtyn and J. B. Lee, The Nielsen numbers of iterations of maps on infra-solvmanifolds of type (R) and periodic points, arXiv:1403.7631.
13 K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathematics, 1639, Springer-Verlag, Berlin, 1996.
14 K. Dekimpe, P. Igodt, S. Kim, and K. B. Lee, Affine structures for closed 3-dimensional manifolds with Nil-geometry, Quart. J. Math Oxford. (2) 46 (1995), no. 182, 141-167.   DOI
15 A. Fel'shtyn and J. B. Lee, The Nielsen and Reidemeister numbers of maps on infrasolvmanifolds of type (R), Topology Appl. 181 (2015), 62-103.   DOI
16 K. Y. Ha and J. B. Lee, Crystallographic groups of Sol, Math. Nachr. 286 (2013), no. 16, 1614-1667.   DOI
17 K. Y. Ha and J. B. Lee, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R)-Corrigendum, Nagoya Math. J. 221 (2016), no. 1, 207-212.   DOI
18 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
19 J. Jezierski, J. Kedra, and W. Marzantowicz, Homotopy minimal periods for NRsolvmanifolds maps, Topology Appl. 144 (2004), no. 1-3, 29-49.   DOI
20 J. Jezierski, Wecken theorem for fixed and periodic points, Handbook of Topological Fixed Point Theory, 555-615, Springer, Dordrecht, 2005.
21 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
22 J. Jezierski and W. Marzantowicz, Homotopy minimal periods for nilmanifold maps, Math. Z. 239 (2002), no. 2, 381-414.   DOI
23 J. Jezierski and W. Marzantowicz, Homotopy minimal periods for maps of three-dimensional nilmanifolds, Pacific J. Math. 209 (2003), no. 1, 85-101.   DOI
24 J. H. Jo and J. B. Lee, Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with Sol4 1-geometry, Fixed Point Theory Appl. 2015 (2015), 175, 1-15.   DOI
25 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.
26 B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Math., 14, Amer. Math. Soc., Providence, R.I., 1983.
27 B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems 4 (1998), no. 2, 301-320.   DOI
28 J. H. Jo and J. B. Lee, Nielsen fixed point theory on infra-solvmanifolds of Sol, Topol. Methods Non-linear Anal. 49 (2017), 325-350.
29 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
30 S. W. Kim and J. B. Lee, Anosov theorem for coincidences on nilmanifolds, Fund. Math. 185 (2005), no. 3, 247-259.   DOI
31 J. B. Lee, Homotopy minimal periods of maps on the Klein bottle, submitted for publication.
32 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
33 J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R), Nagoya Math. J. 196 (2009), 117-134.   DOI
34 J. B. Lee and X. Zhao, Homotopy minimal periods for expanding maps on infranilmanifolds, J. Math. Soc. Japan 59 (2007), no. 1, 179-184.   DOI
35 J. B. Lee and X. Zhao, Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds, Sci. China Ser. A 51 (2008), no. 3, 351-360.   DOI