Linear Fractional Recurrences: Periodicities and Integrability

Eric Bedford; Kyounghee Kim

doi:10.5802/afst.1304

Eric Bedford ¹; Kyounghee Kim ²

¹ Department of Mathematics, Indiana University, Bloomington, IN 47405 USA
² Department of Mathematics, Florida State University, Tallahassee, FL 32306 USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 33-56.

Abstract
Abstract

Linear fractional recurrences are given as $z_{n + k} = A (z) / B (z)$ , where $A (z)$ and $B (z)$ are linear functions of $z_{n}, z_{n + 1}, \dots, z_{n + k - 1}$ . In this article we consider two questions about these recurrences: (1) Find $A (z)$ and $B (z)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$ -step linear fractional recurrences which are periodic of period $4 k$ . The second theorem shows that the Lyness process, $A (z) = a + z_{n + 1} + z_{n + 2} + \dots + z_{n + k - 1}$ and $B (z) = z_{n + 1}$ has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.

Les récurrences fractionnaires linéaires sont données par $z_{n + k} = A (z) / B (z)$ , où $A (z)$ et $B (z)$ sont des fonctions linéaires de $z_{n}, z_{n + 1},$ $\dots, z_{n + k - 1}$ . Dans cet article nous considérons deux questions concernant ces récurrences : (1) Trouver $A (z)$ et $B (z)$ telles que la récurrence soit périodique ; (2) Trouver des intégrales invariantes dans le cas où le degré de l’application birationnelle induite a une croissance quadratique. L’approche de ces questions se fait en considérant l’application birationnelle induite et en déterminant ses degrés dynamiques. Le premier théorème montre que pour tout $k$ il y a des récurrences fractionnaires linéaires à $k$ pas qui sont périodiques de période $4 k$ . Le second théorème montre que le degré du procédé de Lyness $A (z) = a + z_{n + 1} + z_{n + 2} + \dots + z_{n + k - 1}$ et $B (z) = z_{n + 1}$ est à croissance quadratique. Le procédé de Lyness est intégrable, et nous en discuterons les intégrales connues.

MR Zbl

DOI: 10.5802/afst.1304

Author's affiliations:

Eric Bedford ¹; Kyounghee Kim ²

¹ Department of Mathematics, Indiana University, Bloomington, IN 47405 USA
² Department of Mathematics, Florida State University, Tallahassee, FL 32306 USA

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     author = {Eric Bedford and Kyounghee Kim},
     title = {Linear {Fractional} {Recurrences:} {Periodicities} and {Integrability}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {33--56},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Eric Bedford; Kyounghee Kim. Linear Fractional Recurrences: Periodicities and Integrability. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 33-56. doi : 10.5802/afst.1304. https://afst.centre-mersenne.org/articles/10.5802/afst.1304/

References
Cited by

[BK1] Bedford (E.) and Kim K..— Periodicities in linear fractional recurrences: degree growth of birational surface maps. Michigan Math. J. 54, no. 3, p. 647-670 (2006). | MR | Zbl

[BK2] Bedford (E.) and Kim K..— Periodicities in 3-step linear fractional recurrences, preprint.

[CL] Camouzis (E.) and Ladas (G.).— Dynamics of third-order rational difference equations with open problems and conjectures. Advances in Discrete Mathematics and Applications, 5. Chapman & Hall/CRC, Boca Raton, FL, 2008. | MR | Zbl

[CGM1] Cima (A.), Gasull (A.), Mañosa (V.).— Dynamics of some rational discrete dynamical systems via invariants. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16, no. 3, p. 631-645 (2006). | MR | Zbl

[CGM2] Cima (A.), Gasull (A.), Mañosa (V.).— Dynamics of the third order Lyness’ difference equation. J. Difference Equ. Appl. 13, no. 10, p. 855-884 (2007). | MR | Zbl

[CGM3] Cima (A.), Gasull (A.), Mañosa (V.).— Some properties of the $k$ -dimensional Lyness’ map. J. Phys. A 41, no. 28, 285205, 18 pp (2008). | MR | Zbl

[CsLa] Csörnyei (M.) and Laczkovich (M.).— Some periodic and non-periodic recursions, Monatsh. Math. 132, p. 215-236 (2001). | MR | Zbl

[dFE] de Fernex (T.) and Ein (L.).— Resolution of indeterminacy of pairs. Algebraic geometry, p. 165-177, de Gruyter, Berlin (2002). | MR | Zbl

[DF] Diller (J.) and Favre (C.).— Dynamics of bimeromorphic maps of surfaces, Amer. J. of Math., 123, p. 1135-1169 (2001). | MR | Zbl

[GKI] Gao (M.), Kato (Y.) and Ito (M.).— Some invariants for $k$ th-order Lyness equation. Appl. Math. Lett. 17, no. 10, p. 1183-1189 (2004). | MR | Zbl

[GBM] Gardini (L.), Bischi (G.), and Mira (C.).— Invariant curves and focal points in a Lyness iterative process. Dynamical systems and functional equations (Murcia, 2000). Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, no. 7, p. 1841-1852 (2003). | MR | Zbl

[G] Gizatullin (M.).— Rational $G$ -surfaces. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44, no. 1, p. 110-144, 239 (1980). | MR | Zbl

[GL] Grove (E. A.) and Ladas (G.).— Periodicities in nonlinear difference equations, Adv. Discrete Math. Appl., 4, Chapman & Hall and CRC Press, Boca Raton, FL (2005). | MR | Zbl

[H] Hassett (B.).— Introduction to Algebraic Geometry, Cambridge U. Press (2007). | MR | Zbl

[HKY] Hirota (R.), Kimura (K.) and Yahagi (H.).— How to find the conserved quantities of nonlinear discrete equations, J. Phys. A: Math. Gen. 34, p. 10377-10386 (2001). | MR | Zbl

[KL] Kocic (V.I.) and Ladas (G.).— Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers 1993. | MR | Zbl

[KLR] Kocic (V.I.), Ladas (G.), and Rodrigues (I.W.).— On rational recursive sequences, J. Math. Anal. Appl 173, p. 127-157 (1993). | MR | Zbl

[L] Lyness (R. C.).— Notes 1581, 1847, and 2952, Math. Gazette 26 (1942), 62; 29 (1945), 231; 45 (1961), 201.

[M] McMullen (C. T.).— Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes Études Sci. No. 105, p. 49-89 (2007). | Numdam | MR | Zbl

[Z] Zeeman (C.).— Geometric unfolding of a difference equation. Lecture.

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