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A Note on Permutations and Topological Entropy of Continuous Maps of the Interval

Published online by Cambridge University Press:  20 November 2018

Bill Byers
Bill Byers*
Affiliation:
Concordia UniversityMontreal, Quebec, Canada, H4B 1R6
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Abstract

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Suppose f is a continuous endomorphism of an interval which has a periodic orbit, p0 < P1 < … < pn, that defines a permutation a by f(pi) = pσ(i). If σ is irreducible the topological entropy of f is bounded below by the logarithm of the spectral radius of an n x n matrix which is induced by σ.

Keywords

Primary54H20Secondary58F20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 1 , 01 March 1986 , pp. 91 - 95
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Block, L., Simple periodic orbits of mappings of the interval. Trans. Amer. Math. Soc. 254 (1979), pp. 391–398.Google Scholar
2. Block, L. and Hart, D., Stratification of the space of unimodal interval maps, Ergod. Th. and Dynam. Syst. 3 (1983), pp. 533–539.Google Scholar
3. Block, L., Guckenheimer, J., Misiurewicz, M., and Young, L.-S., Periodic points and topological entropy of one dimensional maps, Lect. Notes Math. 819 (Springer 1980), pp. 18–34.Google Scholar
4. Bowen, R. and Franks, J., The periodic points of maps of the disk and the interval, Topology 15 (1976), pp. 337–342.Google Scholar
5. Jonber, L. and Rand, D., The periodic orbit and entropy of certain maps of the unit interval, J. London Math. Soc. (2) 22 (1980), pp. 175–181.Google Scholar
6. Lancaster, P., Theory of Matrices, Academic Press, New York (1969).Google Scholar
7. Misiurewicz, M. and Szlenk, W., Entropy of piecewise monotone mappings, Studia Math. 67 (1980), pp. 45–63.Google Scholar
8. Milnor, J. and Thurston, W., On iterated maps of the interval I and II, Princeton (1977).Google Scholar
9. Stefan, P., A Theorem of Sarkovskii on the coexistence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), pp. 237–248.Google Scholar
10. Takahashi, Y., A formula for topological entropy of one dimensional dynamics, Sci. Papers College Gen. Ed. Univ. Tokyo 30 (1980), pp. 11–22.Google Scholar