Periodic segments and Nielsen numbers

Klaudiusz Wójcik

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 247-252
  • ISSN: 0137-6934

Abstract

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We prove that the Poincaré map φ ( 0 , T ) has at least N ( h ˜ , c l ( W 0 W 0 - ) ) fixed points (whose trajectories are contained inside the segment W) where the homeomorphism h ˜ is given by the segment W.

How to cite

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Wójcik, Klaudiusz. "Periodic segments and Nielsen numbers." Banach Center Publications 47.1 (1999): 247-252. <http://eudml.org/doc/208938>.

@article{Wójcik1999,
abstract = {We prove that the Poincaré map $φ_\{(0,T)\}$ has at least $N(\tilde\{h\}, cl(W_\{0\} \ W_\{0\}^\{-\}) )$ fixed points (whose trajectories are contained inside the segment W) where the homeomorphism $\tilde\{h\}$ is given by the segment W.},
author = {Wójcik, Klaudiusz},
journal = {Banach Center Publications},
keywords = {local process; periodic orbit; Poincaré mapping; Nielsen number; exit set},
language = {eng},
number = {1},
pages = {247-252},
title = {Periodic segments and Nielsen numbers},
url = {http://eudml.org/doc/208938},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Wójcik, Klaudiusz
TI - Periodic segments and Nielsen numbers
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 247
EP - 252
AB - We prove that the Poincaré map $φ_{(0,T)}$ has at least $N(\tilde{h}, cl(W_{0} \ W_{0}^{-}) )$ fixed points (whose trajectories are contained inside the segment W) where the homeomorphism $\tilde{h}$ is given by the segment W.
LA - eng
KW - local process; periodic orbit; Poincaré mapping; Nielsen number; exit set
UR - http://eudml.org/doc/208938
ER -

References

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  1. [1] P. Boyland, Topological methods in surface dynamics, Topol. Appl. 58 (1994), 223-298. Zbl0810.54031
  2. [2] R. F. Brown, On the Nielsen fixed point theorem for compact maps, Duke Math. J. 36 (1969), 699-708. Zbl0186.57002
  3. [3] B. Jiang, Lectures on Nielsen Fixed point Theory, Contemp. Math. vol. 14, AMS Providence, 1983. 
  4. [4] B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems, Contemp. Math. vol. 152, 183-202, AMS Providence 1993 Zbl0798.55001
  5. [5] H. Schirmer, On the location of fixed point sets of pairs of spaces, Topol. and its Appl. 30 (1988), 253-266. 
  6. [6] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986), 65-72. 
  7. [7] H. Schirmer, A Survey of Relative Nielsen Fixed Point Theory, Contemp. Math. vol. 152, 291-309, AMS Providence 1993. Zbl0805.55001
  8. [8] C.C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. no 38, AMS, Providence R.I., 1978, 
  9. [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, Zbl0801.34041
  10. [10] R. Srzednicki, On periodic solutions of planar differential equations with periodic coefficients, J. Diff. Equat. 114 (1994), 77-100, Zbl0811.34031
  11. [11] R. Srzednicki and K. Wójcik, A Geometric Method for Detecting Chaotic Dynamics, J. Diff. Equat. Vol. 135, (1997), 66-82. Zbl0873.58049
  12. [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. Zbl0822.34039
  13. [13] K. Wójcik, Isolating segments and symbolic dynamics, to appear in Nonlinear Anal. Th. Meth. Appl. Zbl0955.37005
  14. [14] X. Z. Zhao, A relative Nielsen number for the complement, Lecture Notes in Math. vol. 1411, Springer-Verlag, 1989, 189-199. 

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