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On the topological entropy of transitive maps of the interval

  • Ethan M. Coven (a1) and Melissa C. Hidalgo (a2)

Abstract

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.

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References

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[1]Barge, M. and Martin, J., ‘Chaos, periodicity, and snake-like continuua’, Trans. Amer. Math. Soc. 289 (1985), 355365.
[2]Barge, M. and Martin, J., ‘Dense periodicity on the interval’, Michigan Math. J. 34 (1987), 311.
[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), 297306.
[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), 1834.
[5]Blokh, A.M., ‘On sensitive mappings of the interval’, Russian Math. Surveys 37 (1982), 203204.
[6]Coppel, W.A., ‘Continuous maps of an interval’, Notes, (Australian National University, 1984).
[7]Coven, E. and Mulvey, I., ‘Transitivity and the center for maps of the circle’, Ergodic Theory Dynamical Systems 6 (1986), 18.
[8]Gantmacher, F., The theory of matrices 2 (Chelsea, New York, 1959).
[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), 865868.
[10]Takahashi, Y., ‘A formula for topological entropy of one-dimensional dynamics’, Sci. Papers College Gen. Ed. Tokyo Univ. 30 (1980), 1122.
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On the topological entropy of transitive maps of the interval

  • Ethan M. Coven (a1) and Melissa C. Hidalgo (a2)

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