Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Grzegorz Graff; Agnieszka Kaczkowska

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2160-2172
  • ISSN: 2391-5455

Abstract

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Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.

How to cite

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Grzegorz Graff, and Agnieszka Kaczkowska. "Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers." Open Mathematics 10.6 (2012): 2160-2172. <http://eudml.org/doc/269429>.

@article{GrzegorzGraff2012,
abstract = {Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.},
author = {Grzegorz Graff, Agnieszka Kaczkowska},
journal = {Open Mathematics},
keywords = {Periodic points; Nielsen number; Fixed point index; Smooth maps; periodic points; fixed point index; smooth maps},
language = {eng},
number = {6},
pages = {2160-2172},
title = {Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers},
url = {http://eudml.org/doc/269429},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Grzegorz Graff
AU - Agnieszka Kaczkowska
TI - Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2160
EP - 2172
AB - Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.
LA - eng
KW - Periodic points; Nielsen number; Fixed point index; Smooth maps; periodic points; fixed point index; smooth maps
UR - http://eudml.org/doc/269429
ER -

References

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  1. [1] Abramowitz M., Stegun I.A. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, 1972 
  2. [2] Apéry R., Irrationalité de ζ(2) et ζ(3), Astérisque, 1979, 61, 11–13 
  3. [3] Batko B., Mrozek M., The Euler-Poincaré characteristic of index maps, Topology Appl., 2007, 154(4), 859–866 http://dx.doi.org/10.1016/j.topol.2006.09.009[WoS][Crossref] Zbl1108.37009
  4. [4] Chow S.-N., Mallet-Parret J., Yorke J.A., A periodic orbit index which is a bifurcation invariant, In: Geometric Dynamics, Rio de Janeiro, July–August, 1981, Lecture Notes in Math., 1007, Springer, Berlin, 1983, 109–131 
  5. [5] Gierzkiewicz A., Wójcik K., Lefschetz sequences and detecting periodic points, Discrete Contin. Dyn. Syst., 2012, 32(1), 81–100 http://dx.doi.org/10.3934/dcds.2012.32.81[WoS][Crossref] Zbl1238.37004
  6. [6] Gompf R.E., The topology of symplectic manifolds, Turkish J. Math., 2001, 25(1), 43–59 Zbl0989.53054
  7. [7] Graff G., Existence of periodic orbits for a perturbed vector field, Topology Proc., 2007, 31(1), 137–143 Zbl1133.37006
  8. [8] Graff G., Jezierski J., Minimal number of periodic points for C 1 self-maps of compact simply-connected manifolds, Forum Math., 2009, 21(3), 491–509 http://dx.doi.org/10.1515/FORUM.2009.023[Crossref][WoS] Zbl1173.37014
  9. [9] Graff G., Jezierski J., Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl., 2011, 158(3), 276–290 http://dx.doi.org/10.1016/j.topol.2010.11.002[Crossref][WoS] Zbl1211.55004
  10. [10] Graff G., Jezierski J., Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds, Discrete Contin. Dyn. Syst. Supplements, 2011, Issue Special, 523–532 Zbl1306.37022
  11. [11] Graff G., Jezierski J., Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply-connected manifolds, J. Fixed Point Theory Appl., 2012, DOI: 10.1007/s11784-012-0076-1 (in press) [WoS][Crossref] Zbl1276.55005
  12. [12] Graff G., Jezierski J., Nowak-Przygodzki P., Fixed point indices of iterated smooth maps in arbitrary dimension, J. Differential Equations, 2011, 251(6), 1526–1548 http://dx.doi.org/10.1016/j.jde.2011.05.024[Crossref] Zbl1247.37020
  13. [13] Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press) Zbl1286.55002
  14. [14] Graff G., Kaczkowska A., Nowak-Przygodzki P., Signerska J., Lefschetz periodic point free self-maps of compact manifolds, Topology Appl., 2012, 159(10–11), 2728–2735 http://dx.doi.org/10.1016/j.topol.2012.03.011[Crossref][WoS] Zbl1263.37038
  15. [15] Heath P.R., A survey of Nielsen periodic point theory (fixed n), In: Nielsen Theory and Reidemeister Torsion, Warsaw, June 24–July 5, 1996, Banach Center Publ., 1999, 49, 159–188 Zbl0943.55002
  16. [16] Jezierski J., Wecken’s theorem for periodic points in dimension at least 3, Topology Appl., 2006, 153(11), 1825–1837 http://dx.doi.org/10.1016/j.topol.2005.06.008[Crossref] Zbl1094.55005
  17. [17] Jezierski J., Marzantowicz W., Homotopy Methods in Topological Fixed and Periodic Points Theory, In: Topol. Fixed Point Theory Appl., 3, Springer, Dordrecht, 2006 Zbl1085.55001
  18. [18] Jiang B.J., Fixed point classes from a differential viewpoint, In: Fixed Point Theory, Sherbrooke, June 2–21, 1980, Lecture Notes in Math., 886, Springer, Berlin-New York, 1981, 163–170 
  19. [19] Jiang B.J., Lectures on Nielsen Fixed Point Theory, Contemp. Math., 14, American Mathematical Society, Providence, 1983 http://dx.doi.org/10.1090/conm/014[Crossref] Zbl0512.55003
  20. [20] Marzantowicz W., Wójcik K., Periodic segment implies infinitely many periodic solutions, Proc. Amer. Math. Soc., 2007, 135(8), 2637–2647 http://dx.doi.org/10.1090/S0002-9939-07-08750-3[Crossref] Zbl1139.37011
  21. [21] Sándor J., Mitrinovic D.S., Crstici B., Handbook of Number Theory I, Springer, Dordrecht, 2006 Zbl1151.11300
  22. [22] Yeates A.R., Hornig G., Dynamical constraints from field line topology in magnetic flux tubes, J. Phys. A, 2011, 44(26), #265501 Zbl1221.85018

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