Jungck theorem for triangular maps and related results

• Grinc, M. ^[2] ; Snoha, L. ^[1]
  1. [1] Matej Bel University

     Matej Bel University

     Eslovaquia

     
  2. [2] Silesian University
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 1, Nº. 1, 2000, págs. 83-92
• Idioma: inglés
• DOI: 10.4995/agt.2000.3025
• Enlaces
  • Texto completo
• Resumen

  • We prove that a continuous triangular map G of the n-dimensional cube In has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.

• Referencias bibliográficas
  • Alsedà Ll., Kolyada S. F. and Snoha L., On topological entropy of triangular maps of the square, Bull. Austral. Math. Soc. 48 (1993), 55–67.
  • Alsedà Ll., Llibre J. and Misiurewicz M., Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publ., Singapore, 1993.
  • Chu S. C. and Moyer R. D., On continuous functions, commuting functions, and fixed points, Fund. Math. 59 (1966), 90–95.
  • Coppel W. A., The solution of equations by iteration, Proc. Cambridge Philos. Soc. 51 (1955), 41–43.
  • Forti G. L., Paganoni L. and Smítal J., Strange triangular maps of the square, Bull. Austral. Math. Soc. 51 (1995), 395–415.
  • Grinč M., On common fixed points of commuting triangular maps, Bull. Polish. Acad. Sci. Math. 47 (1999), 61–67.
  • Jachymski J. R., Equivalent conditions involving common fixed points for maps on the unit interval, Proc. Amer. Math. Soc. 124 (1996), 3229–3233.
  • Jungck G., Common fixed points for commuting and compatible maps on compacta, Proc. Amer. Math. Soc. 103 (1988), 977–983.
  • Jungck G., Common fixed points for compatible maps on the unit interval, Proc. Amer. Math. Soc. 115 (1992), 495–499.
  • Jungck G., Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), 771–779.
  • Kloeden P. E., On Sharkovsky's cycle coexistence ordering, Bull. Austral. Math. Soc. 20 (1979), 171–177.
  • Kolyada S. F., On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems 12 (1992), 749–768.
  • Kolyada S. F. and Snoha L., On ω-limit sets of triangular maps, Real Anal. Exchange 18(1) (1992/93), 115–130.
  • Kolyada S. and Snoha L., Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), 205–233.
  • Martin J. R. and Weiss W., Fixed point sets of metric and nonmetric spaces, Trans. Amer. Math. Soc. 284 (1984), 337–353.
  • Schirmer H., Fixed point sets of continuous selfmaps, Fixed Point Theory Proc. (Sherbrooke, 1980), Lecture Notes in Math., vol. 886, Springer-Verlag,...
  • Sharkovsky A. N., On cycles and the structure of a continuous mapping, Ukrain. Mat. Zh. 17(3) (1965), 104–111 (Russian).
  • Ward L. E., Jr., Fixed point sets, Pacific J. Math. 47 (1973), 553–565.