Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-23T04:03:41.298Z Has data issue: false hasContentIssue false
  • English
  • Français

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†
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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