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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0122-7,
author = {Grzegorz Graff and Agnieszka Kaczkowska},
title = {Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {2160-2172},
zbl = {06137117},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0122-7}
}
Grzegorz Graff; 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, Tome 10 (2012) pp. 2160-2172. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0122-7/
[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] Apéry R., Irrationalité de ζ(2) et ζ(3), Astérisque, 1979, 61, 11–13
[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] | Zbl 1108.37009
[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] 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] | Zbl 1238.37004
[6] Gompf R.E., The topology of symplectic manifolds, Turkish J. Math., 2001, 25(1), 43–59 | Zbl 0989.53054
[7] Graff G., Existence of periodic orbits for a perturbed vector field, Topology Proc., 2007, 31(1), 137–143 | Zbl 1133.37006
[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] | Zbl 1173.37014
[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] | Zbl 1211.55004
[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 | Zbl 1306.37022
[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] | Zbl 1276.55005
[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] | Zbl 1247.37020
[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) | Zbl 1286.55002
[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] | Zbl 1263.37038
[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 | Zbl 0943.55002
[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] | Zbl 1094.55005
[17] Jezierski J., Marzantowicz W., Homotopy Methods in Topological Fixed and Periodic Points Theory, In: Topol. Fixed Point Theory Appl., 3, Springer, Dordrecht, 2006 | Zbl 1085.55001
[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] 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] | Zbl 0512.55003
[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] | Zbl 1139.37011
[21] Sándor J., Mitrinovic D.S., Crstici B., Handbook of Number Theory I, Springer, Dordrecht, 2006 | Zbl 1151.11300
[22] Yeates A.R., Hornig G., Dynamical constraints from field line topology in magnetic flux tubes, J. Phys. A, 2011, 44(26), #265501 | Zbl 1221.85018