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Periodic points for piecewise monotonic transformations

Published online by Cambridge University Press:  19 September 2008

Franz Hofbauer
Franz Hofbauer
Affiliation:
Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
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Abstract

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Periodic points of piecewise monotonic transformations are investigated using an oriented graph, whose closed paths represent the periodic orbits of the transformation. In the first part it is shown that the inverse of the ζ-function is a kind of characteristic polynomial of this graph, which is a power series if the graph is infinite. In the second part, the sets are determined, which can occur as{ with and where T is a monotonic mod one transformation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 2 , June 1985 , pp. 237 - 256
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Block, L., Guckenheimer, J., Misiurewicz, M. & Young, L. S.. Periodic points and topological entropy of one-dimensional maps. Springer Lecture Notes in Mathematics 819, 18–34.CrossRefGoogle Scholar
[2]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J Math. 34 (1979), 213–237. Part II,CrossRefGoogle Scholar
Israel J. Math. 38 (1981), 107–115.CrossRefGoogle Scholar
[3]Hofbauer, F.. Maximal measures for simple piecewise monotonic transformations. Zeitschr. Wahrsch. Verw. Gebiete 52 (1980), 289–300.Google Scholar
[4]Hofbauer, F.. The maximal measure for linear mod one transformations. J. London Math. Soc. 23 (1981), 92–112.Google Scholar
[5]Hofbauer, F.. The structure of piecewise monotonic transformations. Erg. Th. & Dynam. Sys. 1 (1981), 159–178.Google Scholar
[6]Hofbauer, F.. Monotonic mod one transformations. Studia Math. To appear.Google Scholar
[7]Jonker, L. & Rand, D.. Bifurcations in one dimension. Invent. Math. 62 (1981), 347–365.Google Scholar
[8]Milnor, J. & Thurston, W.. On iterated maps of the interval. Preprint, Princeton 1977.Google Scholar