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On the topological entropy of transitive maps of the interval

Published online by Cambridge University Press:  17 April 2009

Ethan M. Coven  and
Melissa C. Hidalgo
Ethan M. Coven
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06457, United States of America
Melissa C. Hidalgo
Affiliation:
Department of Mathematics, University of Hartford, West Hartford, CT 06117, United States of America
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Abstract

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The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 2 , October 1991 , pp. 207 - 213
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Barge, M. and Martin, J., ‘Chaos, periodicity, and snake-like continuua’, Trans. Amer. Math. Soc. 289 (1985), 355–365.CrossRefGoogle Scholar
[2]Barge, M. and Martin, J., ‘Dense periodicity on the interval’, Michigan Math. J. 34 (1987), 3–11.CrossRefGoogle Scholar
[3]Block, L. and Coven, E., ‘Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval’, Trans. Amer. Math. Soc. 300 (1987), 297–306.CrossRefGoogle Scholar
[4]Block, L., Guckenheimer, J., Misiurewicz, M. and Young, L.-S., ‘Periodic points and topological entropy of one-dimensional maps’, Springer Lecture Notes in Math. 819 (1980), 18–34.CrossRefGoogle Scholar
[5]Blokh, A.M., ‘On sensitive mappings of the interval’, Russian Math. Surveys 37 (1982), 203–204.CrossRefGoogle Scholar
[6]Coppel, W.A., ‘Continuous maps of an interval’, Notes, (Australian National University, 1984).Google Scholar
[7]Coven, E. and Mulvey, I., ‘Transitivity and the center for maps of the circle’, Ergodic Theory Dynamical Systems 6 (1986), 1–8.CrossRefGoogle Scholar
[8]Gantmacher, F., The theory of matrices 2 (Chelsea, New York, 1959).Google Scholar
[9]Šarkovskij, A.N., ‘Fixed points and the center of a continuous mapping of the line into itself’, (Ukrainian, Russian and English summaries), Dopovidi Akad. Nauk. Ukrain. RSR (1964), 865–868.Google Scholar
[10]Takahashi, Y., ‘A formula for topological entropy of one-dimensional dynamics’, Sci. Papers College Gen. Ed. Tokyo Univ. 30 (1980), 11–22.Google Scholar