Ukrainian Mathematical Journal

, Volume 56, Issue 8, pp 1242–1257 | Cite as

LI-Yorke sensitivity and other concepts of chaos

  • S. F. Kolyada


We give a survey of the theory of chaos for topological dynamical systems defined by continuous maps on compact metric spaces.


Dynamical System Topological Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Y. Li and J. A. Yorke, “Period three implies chaos,” Amer. Math. Mon., 82, 985–992 (1975).Google Scholar
  2. 2.
    S. Kolyada, On Spatiotemporal Chaos, Preprint 02.26, Max-Planck-Institut für Mathematik, Bonn (2002).Google Scholar
  3. 3.
    E. Akin and S. Kolyada, “Li-Yorke sensitivity,” Nonlinearity, 16, 1421–1433 (2003).Google Scholar
  4. 4.
    L. Alseda, J. Llibre, and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific, River Edge, NJ (1993).Google Scholar
  5. 5.
    L. Block and W. A. Coppel, Dynamics in One Dimension, Lect. Notes Math., 1513, Springer, Berlin (1992).Google Scholar
  6. 6.
    W. de Melo and S. J. van Strien, One-Dimensional Dynamics, Springer, Berlin (1993).Google Scholar
  7. 7.
    A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of One-Dimensional Maps [in Russian], Naukova Dumka, Kiev (1989); English translation: Kluwer, Dordrecht-Boston-London (1997).Google Scholar
  8. 8.
    S. Kolyada and L. Snoha, “Some aspects of topological transitivity—a survey,” Grazer Math. Ber., Bericht., 334, 3–35 (1997).Google Scholar
  9. 9.
    S. Ruette, Chaos for Continuous Interval Maps—a Survey of Relationship between the Various Sorts of Chaos, Preprint (2003) ( Scholar
  10. 10.
    A. N. Sharkovskii, “Coexistence of cycles for continuous maps of the straight line into itself,” Ukr. Mat. Zh., 16, No.1, 61–71 (1964).Google Scholar
  11. 11.
    A. N. Sharkovskii, “On cycles and structure of a continuous map,” Ukr. Mat. Zh., 17, No.1, 104–111 (1965).Google Scholar
  12. 12.
    A. N. Sharkovskii, “Coexistence of cycles of a continuous map of the line into itself,” J. Tolosa. Int. J. Bifurc. Chaos Appl. Sci. Engrg., 5, No.5, 1263–1273 (1995).Google Scholar
  13. 13.
    J. Smítal, “Chaotic functions with zero topological entropy,” Trans. Amer. Math. Soc., 297, 269–282 (1986).Google Scholar
  14. 14.
    V. V. Fedorenko, A. N. Sharkovskii, and J. Smítal, “Characterizations of weakly chaotic maps of the interval,” Proc. Amer. Math. Soc., 110, 141–148 (1990).Google Scholar
  15. 15.
    K. Janková and J. Smítal, “A characterization of chaos,” Bull. Austral. Math. Soc., 34, 283–292 (1986).Google Scholar
  16. 16.
    M. Kuchta and J. Smítal, “Two point scrambled set implies chaos,” in: Eur. Conf. Iteration Theory (ECIT 87), World Scientific, Singapore (1989), pp. 427–430.Google Scholar
  17. 17.
    J. Piorek, “On the generic chaos in dynamical systems,” Univ. Iagel. Acta Math., 25, 293–298 (1985).Google Scholar
  18. 18.
    L. Snoha, “Generic chaos,” Comment. Math. Univ. Carol., 31, 793–810 (1990).Google Scholar
  19. 19.
    L. Snoha, “Dense chaos,” Comment. Math. Univ. Carol., 33, 747–752 (1992).Google Scholar
  20. 20.
    A. M. Bruckner and Hu Thakyin, “On scrambled sets for chaotic functions,” Trans. Amer. Math. Soc., 301, 289–297 (1987).Google Scholar
  21. 21.
    E. Akin, The General Topology of Dynamical Systems, Grad. Stud. Math., Vol. 1, American Mathematical Society, Providence, RI (1993).Google Scholar
  22. 22.
    J. Banks, “Regular periodic decompositions for topologically transitive maps,” Ergod. Theory Dynam. Syst., 17, 505–529 (1997).Google Scholar
  23. 23.
    S. Kinoshita, “On orbits of homeomorphisms,” Colloq. Math., 6, 49–53 (1958).Google Scholar
  24. 24.
    T. Downarowicz and X. Ye, “When every point is either transitive or periodic,” Colloq. Math., 93, 137–150 (2002).Google Scholar
  25. 25.
    J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Stud., Vol. 153, Elsevier, Amsterdam (1988).Google Scholar
  26. 26.
    R. Ellis, “A semigroup associated with a transformation group,” Trans. Amer. Math. Soc., 94, 272–281 (1960).Google Scholar
  27. 27.
    J. Auslander and J. Yorke, “Interval maps, factors of maps and chaos,” Tohoku Math. J., 32, 177–188 (1980).Google Scholar
  28. 28.
    E. Glasner and B. Weiss, “Sensitive dependence on initial conditions,” Nonlinearity, 6, 1067–1075 (1993).Google Scholar
  29. 29.
    E. Akin, J. Auslander, and K. Berg, “When is a transitive map chaotic?” in: Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Vol. 5 (1996), pp. 25–40.Google Scholar
  30. 30.
    R. Adler, A. Konheim, and J. McAndrew, “Topological entropy,” Trans. Amer. Math. Soc., 114, 309–319 (1965).Google Scholar
  31. 31.
    P. Walters, An Introduction to Ergodic Theory, Grad. Texts Math., Vol. 79, Springer (1982).Google Scholar
  32. 32.
    F. Blanchard, B. Host, and A. Maass, “Topological complexity,” Ergodic Theory Dynam. Syst., 20, 641–662 (2000).Google Scholar
  33. 33.
    F. Blanchard, E. Glasner, S. Kolyada, and A. Maass, “On Li-Yorke pairs,” J. Reine Angew. Math., 547, 51–68 (2002).Google Scholar
  34. 34.
    F. Blanchard, B. Host, and S. Ruette, “Asymptotic pairs in positive-entropy systems,” Ergod. Theory Dynam. Syst., 22, 671–686 (2002).Google Scholar
  35. 35.
    D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys., 20, 167–192 (1971); 23, 343–344 (1971).Google Scholar
  36. 36.
    R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, New York (1989).Google Scholar
  37. 37.
    W. Huang and X. Ye, “Devaney’s chaos or 2-scattering implies Li-Yorke chaos,” Topology Appl., 117, 259–272 (2002).Google Scholar
  38. 38.
    W. Gottschalk and G. Hedlund, Topological Dynamics, American Mathematical Society, Providence, RI (1955).Google Scholar
  39. 39.
    J. L. King, “A map with topological minimal self-joinings in the sense of del Junco,” Ergod. Theory Dynam. Syst., 10, 745–761 (1990).Google Scholar
  40. 40.
    H. Furstenberg, “Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, ” Math. Syst. Theory, 1, 1–55 (1967).Google Scholar
  41. 41.
    S. Glasner and D. Maon, “Rigidity in topological dynamics,” Ergod. Theory Dynam. Syst., 9, 309–320 (1989).Google Scholar
  42. 42.
    V. Jiménez López and L. Snoha, “Stroboscopical property in topological dynamics,” Topology Appl., 129, 301–316 (2003).Google Scholar
  43. 43.
    A. Crannell, “A chaotic, non-mixing subshift,” Discrete Contin. Dynam. Systems., 1, 195–202 (1998).Google Scholar
  44. 44.
    E. Akin, Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions, Plenum, New York (1997).Google Scholar
  45. 45.
    S. Kolyada, L. Snoha, and S. Trofimchuk, “Noninvertible minimal maps,” Fund. Math., 168, 141–163 (2001).Google Scholar
  46. 46.
    B. Weiss, “Topological transitivity and ergodic measures,” Math. Syst. Theory, 5, 71–75 (1971).Google Scholar
  47. 47.
    E. Glasner and B. Weiss, “Locally equicontinuous dynamical systems,” Colloq. Math., 84/85, 345–361 (2000).Google Scholar
  48. 48.
    M. Denker, C. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Spaces, Lect. Notes Math., 527, Springer, Berlin (1976).Google Scholar
  49. 49.
    E. Murinova, “Generic chaos in metric spaces,” Acta Univ. M. Belii Ser. Math., 8, 43–50 (2000).Google Scholar
  50. 50.
    E. Akin and E. Glasner, “Residual properties and almost equicontinuity,” J. Anal. Math., 84, 243–286 (2001).Google Scholar
  51. 51.
    A. Iwanik, “Independent sets of transitive points,” Dynam. Syst. Ergod. Theory, 23, 277–282 (1989).Google Scholar
  52. 52.
    J. Mycielski, “Independent sets in topological algebras,” Fund. Math., 55, 139–147 (1964).Google Scholar
  53. 53.
    W. Huang and X. Ye, Dynamical Systems Disjoint from Any Minimal System, Preprint (2003).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • S. F. Kolyada
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyiv

Personalised recommendations