Contingut del Pòster presentat al congrés New Trends in Dynamical SystemsPeer ReviewedPreprin

Bastien, Guy

Mañosa Fernández, Víctor

Rogalski, Marc

English

Contingut del Pòster presentat al congrés New Trends in Dynamical SystemsPeer Reviewe

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Integrable birational maps on the plane: blending dynamics and algebraic geometry

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Integrable birational maps on the plane:blending dynamics and algebraic geometry.Guy Bastien1, V´ıctor Man˜osa2, Marc Rogalski1,31 Universite´ Paris 6 and CNRS, 2 Universitat Polite`cnica de Catalunya (DMA3–CoDALab), 3 Universite´ de Lille 1.IntroductionSome maps arising in mathematical physics are birational planar ones with arational first integral. This means that they preserve a foliation of the planegiven by algebraic curves. As a consequence of Hurwitz theorem if a planarbirational map is not globally periodic then each irreducible invariant curve isof genus 0 or 1.Theorem (Hurwitz)The group of birational maps of a smooth algebraic curve of genus g ≥ 2to itself is finite of order at most 84(g − 1).In the case that the invariant foliation is given (generically) by genus 1 curves(elliptic curves). Then the group structure of the elliptic foliation charac-terizes the dynamics of any birational map preserving it.Theorem ([3])Any birational map F that leaves an elliptic curve E invariant can beexpressed in terms of the group law as either F : P 7→ P + Q orF : P 7→ i(P ) + Q, where i is globally periodic of orders 2,3,4 or 6;and + denotes the inner sum of E .Recall the group law of an elliptic curveObserve that F n(P ) = P + nQ, so E is full of p-periodic orbits iff pQ = 0,that is, iff Q is in the torsion of (E ,+).Example: The rotation number function of2-periodic Lyness recurrences, [1]We consider the 2-periodic Lyness’ equationsun+2 =an + un+1unwhere an =a for n = 2` + 1,b for n = 2`.This equation can studied using the composition map:Fb,a(x, y) := (Fb ◦ Fa)(x, y) =(a + yx,a + bx + yxy)where Fa and Fb are the Lyness maps: Fα(x, y) =(y, α+yx), since(u1, u2) Fa−→ (u2, u3) Fb−→ (u3, u4) Fa−→ (u4, u5) Fb−→ (u5, u6) Fa−→ · · ·Fb,a extends to CP 2 as F˜b,a ([x : y : t]) = [ayt + y2 : at2 + bxt + yt : xy] , andit preserves the foliation given byCh = {(bx + at)(ay + bt)(ax + by + abt)− hxyt = 0}.Some results• For each h s.t. Ch is elliptic, then F˜b,a|Ch(P ) = P + H , where + is thegroup law of Ch taking the infinite point V as the zero element.• If a > 0 and b > 0, then for all h > hc = minQ+(V ), the curves Ch areelliptic.• The rotation number function θb,a(h) is analytic in [hc,+∞)• If (a, b) 6= (1, 1), then ∃ p0(a, b) ∈ N, generically computable, s.t. for anyp > p0(a, b) ∃ at least an oval C+h filled by p–periodic orbits.Again F n(P ) = P + nH, so Ch is full of p-periodic orbits iff pH = V .Following the guidelines of Bastien and Rogalski [2], to study the rotationnumber function we look for a normal form for Ch instead of F˜ :(Ch,+, V )∼=−→ (EL,+, V )F˜|Ch : P 7→ P + H → G|EL : P 7→ P + Ĥwhere EL is the Weierstrass Normal Form:EL = {[x : y : t], y2t = 4x3 − g2 xt2 − g3 t3},Because ∃ ω1 and ω2 depending on a, b and L and a lattice in CΛ = {2nω1 + 2miω2 such that (n,m) ∈ Z2} ⊂ C,such that the Weierstrass ℘ function relative to Λ gives a parametrization ofEL, given byφ : T2 = C/Λ −→ ELz −→ [℘(z) : ℘′(z) : 1] if z /∈ Λ,[0 : 1 : 0] = V if z ∈ Λ.Hence ℘′(z)2 = 4℘(z)3 − g2℘(z)− g3, since y2 = 4x3 − g2x− g3.The oval C+h corresponds with the bounded branch of EL; and [0, ω1] is projectedonto the unbounded semi-branch of EL with negative y–coordinatesIntegrating the differential equation ℘′(z)2 = 4℘(z)3 − g2℘(z)− g3 on [0, u]:u =∫ +∞℘(u)ds√4s3 − g2s− g3 =∫ +∞℘(u)ds√4(s− e1)(s− e2)(s− e3).SinceG|EL : V 7→ V + Ĥ = Ĥ is a rotation of angle Θ(L) ∈[0, 12],and Ĥ has negative ordinate there is a parameter u = 2ω1Θ(L) such thatX(L) := ℘(2ω1Θ(L)), and using that e1 = ℘(ω1), we obtain2Θ(L) =∫ +∞X(L)ds√(s− e1)(s− e2)(s− e3)∫ +∞e1ds√(s− e1)(s− e2)(s− e3).This expression gives the analyticity of the rotation number function and allowsto prove that if (a, b) 6= (1, 1), then the image of θb,a[hc,+∞) is an interval,by studying its asymptotic behavior. 2References[1] G. Bastien, V. Man˜osa, M. Rogalski, On the periodic solutions of 2-periodic Lynessdifference equations. Preprint, arXiv:1201.1027v1 [math.DS][2] G. Bastien, M. Rogalski, Global behavior of the solutions of Lyness’ differenceequation un+2un = un+1 + a, J. Difference Equations and Appl. 10 (2004).[3] D. Jogia, J.A.G. Roberts, F. Vivaldi, An algebraic geometric approach to integrablemaps of the plane, Journal of Physics A 39 (2006).Supported by MCYT grant DPI2011-25822 and Generalitat de Catalunya SGR program.

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Integrable birational maps on the plane: blending dynamics and algebraic geometry

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