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Periodic orbits of difference equations. (English) Zbl 0835.39006

Consider the real difference equation \[ a_{n+2}- (\lambda|a_{n+1} |+ \mu a_{n+1})+ a_n=0 \] and interpret it as a dynamical system \[ \Phi: (a_n, a_{n+1})\to (a_{n+1}, a_{n+2}) \] acting in the plane. Define \(\Lambda_p\) the set of points \((\lambda, \mu)\) for which the mapping \(\Phi\) is periodic. The authors derive some geometric properties of \(\Lambda_p\) (such as unbounded and countable), and they derive criteria for \(\Phi\) to be periodic. The authors also investigate when \(\Phi\) is conjugate to a rotation of the plane, and describes how the rotation numbers of the corresponding circle maps \(\Phi/ |\Phi|\) are related to the structure of \(\Lambda_p\).
For related results, see D. Aharonov and U. Elias [Ergodic Theory Dyn. Syst. 10, No. 2, 209-229 (1990; Zbl 0688.58017)]and A. Brown [Am. Math. Mon. 90, 569 (1983); 92, 218 (1985)].

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations

Citations:

Zbl 0688.58017
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References:

[1] Herman, Asterisque 2 (1986)
[2] DOI: 10.1007/BF02684798 · Zbl 0448.58019 · doi:10.1007/BF02684798
[3] Aharonov, Ergodic Theory Dynamical Systems 10 pp 209– (1990) · Zbl 0702.30008
[4] DOI: 10.2307/2322800 · doi:10.2307/2322800
[5] DOI: 10.2307/3618372 · Zbl 02301497 · doi:10.2307/3618372
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