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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.

Information

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

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