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Periodic orbits of difference equations

Published online by Cambridge University Press:  14 November 2011

A. F. Beardon ,
S. R. Bullett  and
P. J. Rippon
A. F. Beardon
Affiliation:
D.P.M.M.S, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, U.K.
S. R. Bullett
Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London El 4NS, U.K.
P. J. Rippon
Affiliation:
Faculty of Mathematics and Computing, Open University, Milton Keynes MK7 6AA, U.K.
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Abstract

The real difference equation an+2 − (λ|an+1| + μan+1) + an = 0 may be interpreted as a dynamical system Φ:(an, an+1) ↦ (an+1, an+2) acting in the plane. The set ΛP of points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP (for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 4 , 1995 , pp. 657 - 674
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Aharonov, D. and Elias, U.. Invariant curves around a parabolic fixed point at infinity. Ergodic Theory Dynamical Systems 10 (1990), 209–29.CrossRefGoogle Scholar
2Brown, M.. Items in ‘Problems and Solutions’, Amer. Math. Monthly 90 (1983), 569; 92 (1985), 218.CrossRefGoogle Scholar
3Crampin, M.. Piecewise linear recurrence relations. Math. Gaz. 76 (1992), 355–9.CrossRefGoogle Scholar
4Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle. Inst. Hautes Etudes Sci. Publ. Math. 49 (1979), 5–234.CrossRefGoogle Scholar
5Herman, M.. Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau, Vol. 2, Asterisque 144 (Paris: Societe Mathematique de France, 1986).Google Scholar