Beardon, A. F.; Bullett, S. R.; Rippon, P. J. Periodic orbits of difference equations. (English) Zbl 0835.39006 Proc. R. Soc. Edinb., Sect. A 125, No. 4, 657-674 (1995). 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)]. Reviewer: E.Thandapani (Salem) Cited in 7 Documents MSC: 39A12 Discrete version of topics in analysis 39A10 Additive difference equations Keywords:periodic orbits; real difference equation; dynamical system; circle maps Citations:Zbl 0688.58017 PDFBibTeX XMLCite \textit{A. F. Beardon} et al., Proc. R. Soc. Edinb., Sect. A, Math. 125, No. 4, 657--674 (1995; Zbl 0835.39006) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.