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Invariant measures for maps of the interval that do not have points of some period

Published online by Cambridge University Press:  19 September 2008

Jozef Bobok  and
Milan Kuchta
Jozef Bobok
Affiliation:
KM FSv. ČVUT, Thákurova 7, 166 29 Praha 6, Czech Republic
Milan Kuchta
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovak Republic†
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Abstract

We study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 1 , March 1994 , pp. 9 - 21
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

[1]Alsedà, L., Llibre, J. & Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One. Advances Series in Nonlinear Dynamics. Vol. 5, World Scientific: 1993.Google Scholar
[2]Barge, M. & Martin, J.. Dense periodicity on the interval. Proc. Amer. Math. Soc. 94 (1985), 731–735.CrossRefGoogle Scholar
[3]Block, L. & Coven, E. M.. 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]Milnor, J. & Thurston, W.. On iterated maps of the interval. I. The kneading matrix, II. Periodic points. Preprint. Princeton University (1977).Google Scholar
[5]Phelps, R.R.. Lectures on Choquet's Theorem. Van Nostrand, Princeton, NJ, 1966.Google Scholar
[6]Šarkovskiľ, A.N.. Fixed points and the center of a continuous mapping of the line into itself. (Ukrainian; Russian and English summaries.) Dopovidi Akad. Nauk Ukraïn. RSR 7 (1964), 865–868.Google Scholar
[7]Šarkovskiľ, A.N.. Co-existence of cycles of a continuous mapping of the line into itself. (Russian; English summary.) Ukrain. Mat. Ž. 16 (1964), 61–71.Google Scholar
[8]Štefan, P.. A theorem of Šarkovskiľ on the existence of periodic orbits of con tinuous endomorphisms of the real line. Comm. Math. Phys. 54 (1977), 237–248.CrossRefGoogle Scholar