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Asymptotic behaviour of iterated piecewise monotone maps

Published online by Cambridge University Press:  19 September 2008

Jürgen Willms
Jürgen Willms
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, 4800 Bielefeld 1, West Germany
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Abstract

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In this paper the asymptotic behaviour of piecewise monotone functions f: I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f-almost-invariant subsets C1,…, Cr, R1,…, Rs, where each Ci is a disjoint union of closed intervals and each Rj is a Cantor-like subset of I, such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen:

(1) {fn (x)}n ≥ 0 eventually ends up in some Ci.

(2) {fn (x)}n ≥ 0 is attracted to some Rj.

(3) {fn (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fn is monotone and continuous on each connected component of Z.

Moreover, f acts topologically transitively on each Ci and minimally on each Rj. Furthermore, it is shown how the sets C1,…, Cr, R1,…, Rs can be constructed. Finally, our results are applied to some examples.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 1 , March 1988 , pp. 111 - 131
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Barna, B.. Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen, III. Publ. Math. Debrecen 8 (1961), 193–207.Google Scholar
[2]Collet, P. & Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems. Progress in Physics, Vol. 1, Birkhäuser: Boston, 1980.Google Scholar
[3]Collet, P.. Eckmann, J.-P. & Lanford, O. E., III. Universal properties of maps on an interval. Commun. Math. Phys. 76 (1980), 211–254.CrossRefGoogle Scholar
[4]Cornfeld, I. P., Fomin, S. V. & Sinai, Ya. G.. Ergodic Theory. Springer: New York, 1982.CrossRefGoogle Scholar
[5]Coven, E. M. & Nitecki, Z.. Non-wandering sets of the powers of maps of the interval. Ergod. Th. & Dynam. Sys. 1 (1981), 9–31.CrossRefGoogle Scholar
[6]Feigenbaum, M.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.CrossRefGoogle Scholar
[7]Hofbauer, F.. The structure of piecewise monotone transformations. Ergod. Th. & Dynam. Sys. 1 (1981), 135–143.CrossRefGoogle Scholar
[8]Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
[9]Li, T.-Y. & York, J. A.. Period three implies chaos. Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[10]May, R. M.. Biological populations obeying difference equations. J. Theoret. Biol. 51 (1975), 511–524.CrossRefGoogle ScholarPubMed
[11]May, R. M.. Simple mathematical models with very complicated dynamics. Nature 261 (1976), 459–467.CrossRefGoogle ScholarPubMed
[12]Nitecki, Z.. Topological dynamics on the interval. In Ergodic Theory and Dynamical Systems, II. Proc, Special Year Maryland 1979–80. Katok, A., ed., Progress in Math., Vol. 21, Birkhauser: Boston, 1982.Google Scholar
[13]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
[14]Preston, C.. Iterates of Maps on an Interval. Lecture Notes in Mathematics, Vol. 999, Springer: Berlin, 1983.CrossRefGoogle Scholar
[15]Preston, C.. Iterates of piecewise monotone maps on an interval. Preprint, Bielefeld, 01 1984.CrossRefGoogle Scholar
[16]Preston, C., Iterates of piecewise monotone maps on an interval. Preprint, Bielefeld, 1986.Google Scholar
[17]Rényi, A.. Representations for real numbers and their ergodic properties. Acta. Math. Acad. Sci. Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar
[18]Walters, P., Ergodic Theory—Introductory Lectures. Lecture Notes in Mathematics, Vol. 458, Springer: Berlin (1975).Google Scholar
[19]Williams, R. F.. The structure of Lorenz attractors. Publ. Math. I.H.E.S. 50 (1979), 73–99.Google Scholar