Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-28T18:10:16.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Strange triangular maps of the square

Published online by Cambridge University Press:  17 April 2009

G.L. Forti ,
L. Paganoni  and
J. Smítal
G.L. Forti
Affiliation:
Dipartimento de Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italia
L. Paganoni
Affiliation:
Dipartimento de Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italia
J. Smítal
Affiliation:
Institute of Mathematics, Comenius University, 84215 Bratislava, Slovakia and Institute of Mathematics, Silesian University, 74061 Opava, Czech Republic
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.

(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 395 - 415
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Alsedà, L., Kolyada, S.F. and Snoha, L., ‘On topological entropy of triangular maps of the square’, Bull. Austral. Math. Soc. 48 (1993), 55–68.CrossRefGoogle Scholar
[2]Block, L.S. and Coppel, W.A., Dynamics in one dimension, Lectures Notes in Math. 1513 (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar
[3]Bruckner, A.M. and Smítal, J., ‘A characterization of ω-limit sets of maps of the interval with zero topological entropy’, Ergodic Theory Dynamical Systems 13 (1993), 7–19.CrossRefGoogle Scholar
[4]Fedorenko, V.V., Sharkovsky, A.N. and Smítal, J., ‘Characterizations of weakly chaotic maps of the interval’, Proc. Amer. Math. Soc. 110 (1990), 141–148.CrossRefGoogle Scholar
[5]Fedorenko, V.V. and Smítal, J., ‘Maps of the interval Ljapunov stable on the set of non-wandering points’, Acta Math. Univ. Comenian. 50 (1991), 11–14.Google Scholar
[6]Franzová, N. and Smítal, J., ‘Positive sequence topological entropy characterizes chaotic maps’, Proc. Amer. Math. Soc. 112 (1991), 1083–1086.CrossRefGoogle Scholar
[7]Goodman, T.N.T., ‘Topological sequence entropy’, Proc. London Math. Soc. 29 (1974), 331–350.CrossRefGoogle Scholar
[8]Janková, K. and Smítal, J., ‘A characterization of chaos’, Bull. Austral. Math. Soc. 34 (1986), 283–292.CrossRefGoogle Scholar
[9]Kloeden, P.E., ‘On Sharkovsky's cycle coexistence ordering’, Bull. Austr. Math. Soc. 20 (1979), 171–177.CrossRefGoogle Scholar
[10]Kolyada, S.F., ‘On dynamics of triangular maps of the square’, Ergodic Theory Dynamical Systems 12 (1992), 749–768.CrossRefGoogle Scholar
[11]Kuchta, M. and Smiítal, J., ‘Two-point scrambled set implies chaos’, in Proc. Europ. Conf. on Iteration Theory, Caldas de Malavella, Spain 1987 (World Scientific, 1989), pp. 427–430.Google Scholar
[12]Li, T.Y. and Yorke, J., ‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[13]Misiurewicz, M., ‘Invariant measures for continuous transformations of [0, 1] with zero topological entropy’, in Ergodic Theory (Oberwolfach 1978), Lecture Notes in Math. 729 (Springer-Verlag, Berlin, Heidelberg, New York, 1979), pp. 144–152.CrossRefGoogle Scholar
[14]Sharkovsky, A.N., ‘Coexistence of cycles of a continuous map of the line into itself’, (in Russian), Ukrainian Math. J. 16 (1964), 61–71.Google Scholar
[15]Sharkovsky, A.N., Kolyada, S.F., Fedorenko, V.V. and Sivak, A.G., Dynamics of one-dimensional maps, (in Russian) (Naukova Dumka, Kiev, 1989).Google Scholar
[16]Smiítal, J., ‘Chaotic maps with zero topological entropy’, Trans. Amer. Math. Soc. 297 (1986), 269–282.CrossRefGoogle Scholar