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Dynamics of homeomorphisms on minimal sets generated by triangular mappings

Published online by Cambridge University Press:  17 April 2009

Gian Luigi Forti ,
Luigi Paganoni  and
Jaroslav Smítal
Gian Luigi Forti
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy
Luigi Paganoni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy
Jaroslav Smítal
Affiliation:
Institute of Mathematics, Silesian University, 74601 Opava, Czech Republuc
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Abstract

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The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.

To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 1 , February 1999 , pp. 1 - 20
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Block, L.S. and Coppel, W.A., Dynamics in one dimension, Lecture Notes in Math. 1513 (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar
[2]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
[3]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
[4]Forti, G.L. and Paganoni, L., ‘On some properties of triangular maps’, Grazer Math. Ber. (to appear).Google Scholar
[5]Forti, G.L., Paganoni, L. and Smítal, J., ‘Strange triangular maps of the square’, Bull. Austral. Math. Soc 51 (1995), 395–415.CrossRefGoogle Scholar
[6]Kloeden, P.E., ‘On Sharkovsky's cycle coexistence ordering’, Bull. Austral. Math. Soc. 20 (1979), 171–177.CrossRefGoogle Scholar
[7]Kolyada, S.F., ‘On dynamics of triangular maps of the square’, Ergodic Theory Dynamical systems 12 (1992), 749–768.CrossRefGoogle Scholar
[8]Paganoni, L. and Santambrogio, P., ‘On some strange properties of triangular maps’, Grazer Math. Ber. (to appear).Google Scholar
[9]Schweizer, B. and Smítal, J., ‘Measures of chaos and a spectral decomposition of dynamical systems on the interval’, Trans. Amer. Math. Soc. 344 (1994), 737–753.CrossRefGoogle Scholar
[10]Schweizer, B., Sklar, A. and Smítal, J., ‘Distributional (and other) chaos and its measurement’, (preprint).Google Scholar
[11]Sibirsky, K.S., Introduction to topological dynamics (Noordhoff International Publishers, Leiden, 1975).Google Scholar