# American Institute of Mathematical Sciences

February  2012, 32(2): 587-604. doi: 10.3934/dcds.2012.32.587

## Rational periodic sequences for the Lyness recurrence

 1 Dept. de Matemµatiques, Universitat Autónoma de Barcelona, Edifici C, 08193-Bellaterra, Barcelona, Spain 2 Dept. de Matemàtica Aplicada III (MA3), Control, Dynamics and Applications Group (CoDALab), Universitat Politècnica de Catalunya (UPC), Colom 1, 08222 Terrassa, Spain 3 Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Ediﬁci C, 08193 Bellaterra, Barcelona, Spain

Received  September 2010 Revised  December 2010 Published  September 2011

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\mathbb{Q}$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves.
Citation: A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587
##### References:
 [1] A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization,, Math. Comp., 60 (1993), 399.  doi: 10.1090/S0025-5718-1993-1140645-1.  Google Scholar [2] E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion,, J. Difference Equations Appl., 1 (1995), 291.  doi: 10.1080/10236199508808028.  Google Scholar [3] G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$,, J. Difference Equations Appl., 10 (2004), 977.  doi: 10.1080/10236190410001728104.  Google Scholar [4] A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières,, C. R. Acad. Sci. Paris, 294 (1982), 657.   Google Scholar [5] F. Beukers and R. Cushman, Zeeman's monotonicity conjecture,, J. Differential Equations, 143 (1998), 191.  doi: 10.1006/jdeq.1997.3359.  Google Scholar [6] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar [7] H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations,", Graduate Texts in Mathematics, 239 (2007).   Google Scholar [8] J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().   Google Scholar [9] A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().   Google Scholar [10] N. Elkies, Rational points near curves and small nonzero $|x^3 - y^2|$ via lattice reduction,, in, 1838 (2000), 33.   Google Scholar [11] J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding,, Discrete Comput. Geom., 25 (2001), 477.  doi: 10.1007/s004540010075.  Google Scholar [12] D. Husemoller, "Elliptic Curves,", With an appendix by Ruth Lawrence, 111 (1987).   Google Scholar [13] D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane,, J. Phys. A, 39 (2006), 1133.  doi: 10.1088/0305-4470/39/5/008.  Google Scholar [14] I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'', Fifth edition, (1991).   Google Scholar [15] F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'', Thèse de doctorat, (2008).   Google Scholar [16] W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().   Google Scholar [17] J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'', Graduate Texts in Mathematics, 151 (1994).   Google Scholar [18] J. Silverman, "The Arithmetic of Elliptic Curves,'', Second edition, 106 (2009).   Google Scholar [19] J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'', Undergraduate Texts in Mathematics, (1992).   Google Scholar [20] J. M. H. Olmsted, Rational values of trigonometric functions,, Amer. Math. Monthly, 52 (1945), 507.  doi: 10.2307/2304540.  Google Scholar [21] E. C. Zeeman, Geometric unfolding of a difference equation,, Hertford College, (1996).   Google Scholar

show all references

##### References:
 [1] A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization,, Math. Comp., 60 (1993), 399.  doi: 10.1090/S0025-5718-1993-1140645-1.  Google Scholar [2] E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion,, J. Difference Equations Appl., 1 (1995), 291.  doi: 10.1080/10236199508808028.  Google Scholar [3] G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$,, J. Difference Equations Appl., 10 (2004), 977.  doi: 10.1080/10236190410001728104.  Google Scholar [4] A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières,, C. R. Acad. Sci. Paris, 294 (1982), 657.   Google Scholar [5] F. Beukers and R. Cushman, Zeeman's monotonicity conjecture,, J. Differential Equations, 143 (1998), 191.  doi: 10.1006/jdeq.1997.3359.  Google Scholar [6] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar [7] H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations,", Graduate Texts in Mathematics, 239 (2007).   Google Scholar [8] J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().   Google Scholar [9] A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().   Google Scholar [10] N. Elkies, Rational points near curves and small nonzero $|x^3 - y^2|$ via lattice reduction,, in, 1838 (2000), 33.   Google Scholar [11] J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding,, Discrete Comput. Geom., 25 (2001), 477.  doi: 10.1007/s004540010075.  Google Scholar [12] D. Husemoller, "Elliptic Curves,", With an appendix by Ruth Lawrence, 111 (1987).   Google Scholar [13] D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane,, J. Phys. A, 39 (2006), 1133.  doi: 10.1088/0305-4470/39/5/008.  Google Scholar [14] I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'', Fifth edition, (1991).   Google Scholar [15] F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'', Thèse de doctorat, (2008).   Google Scholar [16] W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().   Google Scholar [17] J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'', Graduate Texts in Mathematics, 151 (1994).   Google Scholar [18] J. Silverman, "The Arithmetic of Elliptic Curves,'', Second edition, 106 (2009).   Google Scholar [19] J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'', Undergraduate Texts in Mathematics, (1992).   Google Scholar [20] J. M. H. Olmsted, Rational values of trigonometric functions,, Amer. Math. Monthly, 52 (1945), 507.  doi: 10.2307/2304540.  Google Scholar [21] E. C. Zeeman, Geometric unfolding of a difference equation,, Hertford College, (1996).   Google Scholar
 [1] David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335 [2] Joseph H. Silverman. Local-global aspects of (hyper)elliptic curves over (in)finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 101-114. doi: 10.3934/amc.2010.4.101 [3] Ferruh Özbudak, Burcu Gülmez Temür, Oǧuz Yayla. Further results on fibre products of Kummer covers and curves with many points over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 151-162. doi: 10.3934/amc.2016.10.151 [4] Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 1-10. doi: 10.3934/amc.2011.5.1 [5] Alice Silverberg. Some remarks on primality proving and elliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 427-436. doi: 10.3934/amc.2014.8.427 [6] Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327 [7] Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289-292. doi: 10.3934/amc.2017020 [8] Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010 [9] Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 53-59. [10] K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62. [11] Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171 [12] Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009 [13] Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107 [14] Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215 [15] Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure & Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601 [16] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [17] Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525 [18] Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81 [19] Leonardo Manuel Cabrer, Daniele Mundici. Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4723-4738. doi: 10.3934/dcds.2016005 [20] Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583

2018 Impact Factor: 1.143