Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Calculation of Lefschetz and Nielsen numbers in hyperspaces for fractals and dynamical systems


Authors: J. Andres and M. Väth
Journal: Proc. Amer. Math. Soc. 135 (2007), 479-487
MSC (2000): Primary 37B99; Secondary 47H04, 47H09, 47H10, 54H25.
DOI: https://doi.org/10.1090/S0002-9939-06-08505-4
Published electronically: August 10, 2006
MathSciNet review: 2255294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A simple argument is given as to why it is always trivial to calculate Lefschetz and Nielsen numbers for iterated function systems or dynamical systems in hyperspaces. The problem is reduced to a simple combinatorical situation on a finite set.


References [Enhancements On Off] (What's this?)

  • 1. Andres, J., Continuation principles for fractals, Fixed Point Theory 5 (2004), no. 2, 165-180. MR 2117330 (2005m:28017)
  • 2. -, Applicable fixed point principles, Handbook of Topological Fixed Point Theory (Brown, R. F., Furi, M., Górniewicz, L., and Jiang, B., eds.), Springer, Berlin, 2005, 687-739. MR 2170491
  • 3. Andres, J. and Fišer, J., Metric and topological multivalued fractals, Int. J. Bifurc. Chaos 14 (2004), no. 4, 1277-1289. MR 2063892 (2005c:28011)
  • 4. Andres, J., Fišer, J., Gabor, G., and Lesniak, K., Multivalued fractals, Chaos Solitons Fractals 24 (2005), no. 3, 665-700. MR 2116280 (2005h:28015)
  • 5. Andres, J. and Górniewicz, L., Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. MR 1998968 (2005a:47102)
  • 6. Andres, J. and Väth, M., Two topological definitions of a Nielsen number for coincidences of noncompact maps, Fixed Point Theory Appl. 2004 (2004), no. 1, 49-69. MR 2058235 (2005e:55001)
  • 7. Arens, R. F. and Eells, James, J., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397-403. MR 0081458 (18:406e)
  • 8. Curtis, D. W., Hyperspaces of noncompact metric spaces, Compositio Math. 40 (1980), no. 2, 139-152. MR 0563538 (81c:54009)
  • 9. Dugundji, J., Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Compositio Math. 13 (1958), 229-246. MR 0113217 (22:4055)
  • 10. Granas, A., Generalizing the Hopf-Lefschetz fixed point theorem for non-compact ANRs, Symposium on Infinite Dimension Topology, Bâton-Rouge, 1967, Ann. Math. Studies, vol. 27, 1972, 119-130. MR 0410729 (53:14475)
  • 11. Kelley, J. L., Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 0006505 (3:315b)
  • 12. Lesniak, K., Fixed points of the Barnsley-Hutchinson operators induced by hyper-condensing maps, Le Matematiche (to appear).
  • 13. -, Extremal sets as fractals, Nonlinear Anal. Forum 7 (2002), no. 2, 199-208. MR 1959879 (2004a:54020)
  • 14. Michael, E., Topologies of spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 (13:54f)
  • 15. Nussbaum, R. D., Generalizing the fixed point index, Math. Ann. 228 (1977), 259-278. MR 0440587 (55:13461)
  • 16. Ok, E. A., Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal. 56 (2004), 309-330. MR 2032033 (2004k:47112)
  • 17. Ruiz del Portal, F. R. and Salazar, J. M., Fixed point index in hyperspaces: A Conley-type index for discrete semidynamical systems, J. London Math. Soc. (2) 64 (2001), 191-204. MR 1840779 (2002e:54023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37B99, 47H04, 47H09, 47H10, 54H25.

Retrieve articles in all journals with MSC (2000): 37B99, 47H04, 47H09, 47H10, 54H25.


Additional Information

J. Andres
Affiliation: Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic
Email: andres@inf.upol.cz

M. Väth
Affiliation: Department of Mathematics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany
Address at time of publication: Freie Universität Berlin, Fachbereich Mathematik und Informatik (WE1), Sekretariat Prof. B. Fiedler, Arnimallee 2-6, 14195 Berlin, Germany
Email: vaeth@mathematik.uni-wuerzburg.de

DOI: https://doi.org/10.1090/S0002-9939-06-08505-4
Keywords: Multivalued fractal, iterated function system, hyperspace, Lefschetz number, Nielsen number, ultimate range, condensing map, ANR
Received by editor(s): March 24, 2005
Received by editor(s) in revised form: September 20, 2005
Published electronically: August 10, 2006
Additional Notes: The first author was supported by the Council of Czech Government (MSM 6198959214)
This paper was written while the second author was a Heisenberg fellow of the DFG (Az. VA 206/1-1) and invited by the University of Olomouc. Financial support by the DFG is gratefully acknowledged.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society