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Seminari "Birational maps on elliptic curves: blending dynamics and algebraic geometry", a càrrec de Víctor Mañosa (UPC)
28/03/2012 09:00
29/03/2012 09:00

Birational planar maps possessing a rational first integral, preserve a foliation of the plane given by algebraic curves. We will review some results which state that the most typical situation is that this algebraic foliation will be given either by conics and straight lines or by elliptic curves. In the last case we will see some nice results showing that the group structure of the elliptic foliation characterizes the dynamics of any birational map preserving it. All these results are classical and well-known in algebraic geometry, see [2] and [3]. This will be the main core of the talk, and it is aimed to be expository and addressed to a general audience.
To exemplify the above stuff we will see how it works on the Lyness map F(x,y)=(y,(a+y)/x). This map preserves an algebraic foliation given by curves which are, generically, elliptic. We will see how on each of these elliptic curves the map is an affine action in terms of the group structure of the curve. In fact, we will see that the Lyness’ one is a universal family of elliptic curves.
Finally we will review the group structure of rational elliptic curves and its relation with the existence of rational periodic orbits; we will do a brief digression on numerical experiments; and will give a negative answer to a conjecture of Zeeman (and an open problem of Bastien an Rogalski) about the existence of rational 9-periodic orbits of the Lyness map, see [1,4] and [5].

[1] G. Bastien, M. Rogalski, Global behavior of the solutions of Lyness' difference equation u_{n+2}u_n=u_{n+1}+a, J. Difference Equations and Appl. 10 (2004), 977-1003.
[2] J. Duistermaat. Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer-Verlag, 2010.
[3] D. Jogia. J.A.G. Roberts, F. Vivaldi, An algebraic geometric approach to integrable maps of the plane. Journal of Physics A, 39 (2006), 1133--1149.
[4] A. Gasull, V. Mañosa, X. Xarles. Rational Periodic Sequences for the Lyness Equation. Discrete and Continuous Dynamical Systems -series A. 32 (2012), 587-604.
[5] E.C. Zeeman. Geometric unfolding of a difference equation, Preprint Hertford College, Oxford (1996). Unpublished.