We prove that the Poincaré map has at least fixed points (whose trajectories are contained inside the segment W) where the homeomorphism is given by the segment W.
@article{bwmeta1.element.bwnjournal-article-bcpv47i1p247bwm, author = {W\'ojcik, Klaudiusz}, title = {Periodic segments and Nielsen numbers}, journal = {Banach Center Publications}, volume = {50}, year = {1999}, pages = {247-252}, zbl = {0940.55003}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p247bwm} }
Wójcik, Klaudiusz. Periodic segments and Nielsen numbers. Banach Center Publications, Tome 50 (1999) pp. 247-252. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p247bwm/
[000] [1] P. Boyland, Topological methods in surface dynamics, Topol. Appl. 58 (1994), 223-298. | Zbl 0810.54031
[001] [2] R. F. Brown, On the Nielsen fixed point theorem for compact maps, Duke Math. J. 36 (1969), 699-708. | Zbl 0186.57002
[002] [3] B. Jiang, Lectures on Nielsen Fixed point Theory, Contemp. Math. vol. 14, AMS Providence, 1983.
[003] [4] B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems, Contemp. Math. vol. 152, 183-202, AMS Providence 1993 | Zbl 0798.55001
[004] [5] H. Schirmer, On the location of fixed point sets of pairs of spaces, Topol. and its Appl. 30 (1988), 253-266.
[005] [6] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986), 65-72.
[006] [7] H. Schirmer, A Survey of Relative Nielsen Fixed Point Theory, Contemp. Math. vol. 152, 291-309, AMS Providence 1993. | Zbl 0805.55001
[007] [8] C.C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. no 38, AMS, Providence R.I., 1978,
[008] [9] R. Srzednicki, Periodic and bounded solutions in block for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. Theory Meth. Appl. 22 no 6 (1994), 707-737, | Zbl 0801.34041
[009] [10] R. Srzednicki, On periodic solutions of planar differential equations with periodic coefficients, J. Diff. Equat. 114 (1994), 77-100, | Zbl 0811.34031
[010] [11] R. Srzednicki and K. Wójcik, A Geometric Method for Detecting Chaotic Dynamics, J. Diff. Equat. Vol. 135, (1997), 66-82. | Zbl 0873.58049
[011] [12] R. Srzednicki, A Geometric Method for the Periodic Problem in Ordinary Differential Equations, Seminaire D'Analyse Moderne No.22, Eds.: G. Fournier, T. Kaczynski, Université de Sherbrooke 1992. | Zbl 0822.34039
[012] [13] K. Wójcik, Isolating segments and symbolic dynamics, to appear in Nonlinear Anal. Th. Meth. Appl. | Zbl 0955.37005
[013] [14] X. Z. Zhao, A relative Nielsen number for the complement, Lecture Notes in Math. vol. 1411, Springer-Verlag, 1989, 189-199.