ON THE ALGEBRAIC DIFFERENCE EQUATIONS un

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u

n+2

u

n=

ψ(u

n+1

) IN

_R⁺_∗

, RELATED TO A FAMILY OF ELLIPTIC QUARTICS IN THE PLANE

G. BASTIEN AND M. ROGALSKI

Received 20 October 2004 and in revised form 27 January 2005

We continue the study of algebraic diﬀerence equations of the typeun+2un=ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quarticsQ(K) of the plane. We prove, as in “on some algebraic diﬀerence equationsun+2un=ψ(un+1) inR⁺_∗, related to families of conics or cubics:

generalization of the Lyness’ sequences” (2004), that the solutionsMn=(un+1,un) are persistent and bounded, move on the positive componentQ⁰(K) of the quarticQ(K) which passes throughM0, and diverge ifM0is not the equilibrium, which is locally stable. In fact, we study the dynamical systemF(x,y)=((a+ bx+cx²)/ y(c+dx+x²),x), (a,b,c,d)∈R⁺⁴,a+b >0,b+c+d >0, inR⁺_∗², and show that its restriction toQ⁰(K) is conjugated to a rotation on the circle. We give the possible periods of solutions, and study their global behavior, such as the density of initial periodic points, the density of trajecto- ries in some curves, and a form of sensitivity to initial conditions. We prove a dichotomy between a form of pointwise chaotic behavior and the existence of a common minimal period to all nonconstant orbits ofF.

1. Introduction

In [4], we study the diﬀerence equations

un+2un=a+bun+1+u²_n+1, un+2un=a+bun+1+cu²_n+1

c+u_n+1 (1.1)

which generalize the Lyness’ diﬀerence equationsu_n+2u_n=a+u_n+1(see [2,7,8,9]). The first of these equations is related to a family of conics, and the second to a family of cubics (whose Lyness’ cubics are particular cases). The results of [4] in the two cases are analogous to the results obtained in [3] about the global behavior of the solutions of Lyness’ diﬀerence equation.

In the present paper, we will study the diﬀerence equation un+2un=a+bun+1+cu²_n+1

c+dun+1+u²_n+1 . (1.2)

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The dynamical system inR⁺_∗²which represents this diﬀerence equation is F(x,y)=

a+bx+cx² yc+dx+x²,x

. (1.3)

It is well defined as a homeomorphism ofR⁺_∗²whena,b,c,d≥0 anda+b+c >0, as we always assume. We have

Mn+1=

un+2,un+1

=FMn

=Fun+1,un

. (1.4)

There is an invariant function

G(x,y)=xy+d(x+y) +c x

y+ y x

+b 1

x+1 y

+ a

xy, (1.5)

which satisfiesG◦F=G, and thusG(un+1,un) is constant on every solution of (1.2).

IfK=G(u1,u0), the quarticQ(K) with equationG(x,y)=K, or

x²y²+dxy(x+y) +cx²+y²+b(x+y) +a−Kxy=0, (1.6) passes throughM0.

The quarticsQ(K) are invariant on the action ofF, and thus the pointsM_nmove on the quartic passing throughM0, more precisely on its positive componentQ⁰(K).

The mapFhas a geometrical interpretation. IfM∈R⁺_∗², letMbe the second point of the quarticQ(K) which passes throughMwhose first coordinate is the same as those of M(there is only one such pointMbecause the point at the vertical infinity is a double point of the quartic). The imageF(M) is the symmetric point ofMwith respect to the diagonalx₌y.

For all this results, we refer to [4].

InSection 2, we give a general topological result useful for our study, which extends a result of [4], and we define a general property of weak chaotic behavior, whose proof for (1.2) is the goal of this paper.

InSection 3, we use this result to show that the solutions of diﬀerence equation (1.2) are, ifa+b >0 andb+c+d >0, bounded and persistent inR⁺_∗², and diverge if (u1,u0)= (,), the fixed point ofF, and prove that this point is locally stable.

InSection 4, we show that the case whereun+2unis a homographic function ofun+1, studied in [5], comes down to our general model (1.2). This gives again, in a simpler way, results of [5], and improvements of them.

InSection 5, we study the casea=0, where the quartic passes through the point (0, 0).

This case is easy, because a simple birational map transforms every quarticQ(K) into a cubic curve studied in [4]. So we can apply the results of [4] without more work.

InSection 6, we prove general results in the casea >0, which lead to the fact that the restriction of the mapFto each curveQ⁰(K) is conjugated to a rotation onto the circle (seeTheorem 6.11). We study also in Sections6and7whether the chaotic behavior defined inSection 2holds in the general case of (1.2), with a general property of dichotomy (seeTheorem 6.18), and what happens in some particular cases (Section 7) and in the general one (Section 8).

InSection 9, we determine the possible periods of solutions of (1.2).

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2. A topological tool for diﬀerence equations with an invariant

In this section, we give an abstract and more or less classical general result which will be useful for the study of diﬀerence equations. This assertion extends [4, Proposition 1].

Proposition2.1. LetXbe a topological Hausdorﬀspace. LetF:X→XandG:X→Rbe two maps. Suppose first that the following conditions hold:

(a)Fis continuous onX;

(b)Gis continuous and has a strict minimumK_mat a pointL;

(c)∀x∈X,G◦F(x)=G(x)(the invariance property);

(d)Fhas at most one fixed point.

IfK≥K_m, the level sets (if nonempty) ofG are defined byᏯK= {x∈X|G(x)=K}. Then the following three results hold:

(1)every pointx∈Xlies in exactly one setᏯK; (2)the pointLis the (unique) fixed point ofF;

(3)ifM0∈X let Mn+1=F(Mn)be the points of the orbit ofM0 underF; thenMn∈ ᏯG(M0), and ifM0=L, then the sequence(Mn)does not converge.

Now suppose additional hypotheses:

(e)Xis connected and locally compact;

(f)K_∞:=limx→∞G(x)≤+∞exists, andG < K_∞; then

(4)each ᏯK is compact and nonempty for K_m≤K < K_∞ (withᏯKm= {L}), and the equilibrium pointLis locally stable.

Suppose at last the additional hypothesis:

(g)Ghas only one local minimum (its global one atL); then

(5)for K > Km the set ᏯK is the boundary of the open set UK = {G < K}which is a connected relatively compact set.

Proof. Assertions (1) and (2) are obvious. IfMn+1=F(Mn), thenMn∈ᏯG(M0). Suppose thatM_nconverges to a pointN. ThenG(N)₌G(M0) andF(N)₌N, so by (d) and (1) N=L. ButG(M_n)=G(N)=G(L)=K_m, and by (b)M_n=N for alln. Thus, ifM0=L, thenMndoes not converge.

If (e) and (f) hold, it is easy to see thatᏯKis nonempty and compact for everyK≥Km; in particular, sequences (M_n) are bounded (i.e., relatively compact).

We prove now that the setsUK= {G < K}form a basis of neighborhoods ofL. LetV be an open neighborhood ofL. The sets{G≤K}, forK > Km, are compact, and their intersection is{L}; so there is aK > K_msuch that{G≤K} ⊂V, and thusU_K⊂V.

We can now prove easily thatLis locally stable: ifVis a neighborhood ofL, there exists K > Kmsuch thatUK⊂V. IfM0∈UK, then, for everyn,Mn∈UK by (c), andMn∈V.

We prove now assertion (5), if (g) also holds. We haveU_K⊂ {G≤K}, andU_K\U_K⊂ {G≤K} \ {G < K} =ᏯK. Thus,∂UK⊂ᏯK. Now, ifᏯK⊂∂UK, there existsx∈ᏯK,x /∈

∂UK, thus there exists a neighborhoodV ofxsuch thatV∩UK = ∅. Thus,G≥K on V, andG(x)=K; thus x is a local minimum ofG, andx=LbecauseK > K_m: this is impossible, and∂UK=ᏯK.

Finally, we prove thatUK is connected. IfUK is the union of two disjoint nonempty open setsAandB(which are relatively compact), putα=infAGandβ=infBG; we have α,β < K. Ifα=G(u) withu∈Aandβ=G(v) withv∈B, thenuandvare two distinct

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local minima ofG, which is impossible. Thus we can suppose thatα=G(u) withu∈ A\A. ButA⊂X\B(becauseU_K is open), thusu /∈B, andu∈U_K\U_K=∂U_K=ᏯK. So

we haveG(u)=K > α, which is a contradiction.

We will useProposition 2.1whenXis an open subset ofR²; in this context,FandG are given by (2) and (3). In the general case, we can ask the question whether a form of chaotic behavior may be described for the mapF(we will study in this paper if it is the case withFandGgiven by (2) and (3)). Precisely, one may ask the question whetherF has an “invariant pointwise chaotic behavior,” denoted IPCB.

Property of IPCB. We suppose that X, F, and G satisfy properties (a), (b),. . ., (g) of Proposition 2.1, and thatXis a metric space, with distanced. We say that the dynamical system (X,F) with invariantGhas IPCB if we have the following three properties.

(a) There exists a partition ofX\{L}into two dense subsetsAandBwhich both are union of “curves”ᏯK, and then invariant underF:Ais the set of initial periodic pointsM0,Bis the set of initial pointsM0whose orbit is dense in the curveᏯK

which passes throughM0(that isᏯG(M0)).

(b) Every pointM0∈X\{L}has sensitivity to initial conditions, that is, there exists δ(M0)>0 (this dependance onM0explains the term “pointwise”) such that every neighborhood ofM0contains a pointM0whose iteratesM_n satisfyd(Mn,M_n)≥ δ(M0) for infinitely many integersn.

(c) There exists an integerNsuch that every integern≥Nis the minimal period of some periodic orbit ofF.

IPCB is the essential result of [3] about the behavior of Lyness’ diﬀerence equation un+2un=k+un+1if 0< k=1 (ifk=1, 5 is a common minimal period to all nonconstant solutions).

In [4], we prove also that IPCB holds for the solutions of diﬀerence equations inR⁺_∗² un+2un=a+bun+1+u²_n+1, un+2un=a+bun+1+cu²_n+1

c+u_n+1 . (2.1)

An important tool to study the dynamical system linked to (1.2) may be an eventual property in the abstract case ofProposition 2.1: for everyK∈]K_m,K_∞[, is the dynamical systemF_|_Ꮿ_K conjugated to a rotation on the circle with angle 2πθ(K)∈]0,π[? This even- tual property supposes that each setᏯK is homeomorphic to a circle. Then the study of the properties of functionθwould be essential: continuity (analyticity ifXis an open set ofR²), limits whenK→KmandK→K_∞.

3. First general results of divergence and stability We begin by identifying the fixed point.

Lemma3.1. Ifa₌b₌0, then sequence (1.2) tends to0. Ifa+b >0, then the fixed point of the dynamical system (1.3) is the unique positive rootof the equation

Y⁴+dY³−bY−a=0, (3.1)

and it is the unique possible limit for sequence (1.2).

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Figure 3.1

Proof. It is obvious that a fixed point ofFhas the form (Y,Y) whereYsatisfies (3.1), and that (3.1) has a unique positive rootsuch that (,) is invariant byF.

For the limit of the sequence (un) of (1.2), we must be more careful ifa=0, because in this caseY=0 is solution of (3.1). But thenQ(K) passes through (0, 0) and has as tangent at this point the linex+y=0 ifb >0, and so the pointM_n=(u_n+1,u_n), which lies onQ(K)∩R⁺_∗², cannot tend to (0, 0).

Ifa=b=0, the fixed point is solution of Y⁴+dY³=0, which has no solution in R⁺_∗. In this case, we have un+2/un+1=(un+1/un)(c/(c+dun+1+u²_n+1)), with c >0 and d≥0. Thus ρn=un+1/unis decreasing and tends to a limitλ. Ifλ <1, thenun→0. If λwould satisfyλ≥1, thenu_nwould be increasing, and would tend to infinity. But then c/(c+dun+1+u²_n+1)≤1/2 for bign, and thus we would haveλ=0, which is a contradic-

tion.

With the objective of usingProposition 2.1, it is necessary to study the functionG. The first question is to know ifG(x,y)→+∞if (x,y) tends to the infinite point of the locally compact spaceR⁺_∗². It appears that this condition fails in the general case. Indeed, we look for a condition for the setsA_K:= {G≤K} ∩R⁺_∗²to be compact. The hypothesis is

xy+d(x+y) +c x

y+ y x

+b 1

x+1 y

+ a

xy ^≤K. (3.2)

Thus we havexy+a/xy≤K,d(x+y)≤K,c(x/ y+y/x)≤K,b/x≤K, andb/ y≤K. Ifb >0, thenx≥b/Kandy≥b/K, and thus, with the conditionxy≤K, the setA_K is compact. Ifb=0, we will supposea >0 (the casea=b=0 is trivial byLemma 3.1), and the conditionxy+a/xy≤Kimplies that 0< r1≤xy≤r2: the point (x,y) is between two hyperbolas. But then ifcordis positive, we havex/ y+y/x≤K/cand thus 0< s1≤y/x≤ s2, orx+y≤K/b. In the two cases,A_K is compact; seeFigure 3.1.

So the good condition in (1.2), which we suppose in all the sequel, is

a+b >0, b+c+d >0. (3.3) It is to be noticed that ifb=c=d=0, the function Gdoes not tend to +∞at the point at infinity ofR⁺_∗², and then the solutions of the diﬀerence equation (1.2) may be unbounded or not persistent. In fact, it is always the case.

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Indeed, consider the diﬀerence equationu_n+2u_n=a/u²_n+1, witha >0. Its solutions are the sequencesu_n=a^1/4exp[(−1)ⁿ(A+Bn)], withA=ln(u0a⁻^1/4) andB=−ln(u0u1a⁻^1/2), which are neither bounded nor persistent ifu0u1=√

a.

Then, we must identify the minimum ofG. The equations of critical points arex²y²+ dx²y+c(x²−y²)−by−a=0 andx²y²+d y²x+c(y²−x²)−bx−a=0. The diﬀerence of these two equations gives (x−y)(dxy+ 2c(x+y) +b)=0. But if (x,y)∈R⁺_∗², the only solution isx=y, so the previous equations givex⁴+dx³−bx−a=0, and thus we have x=y=:Ghas a unique critical point at the equilibriumL=(,), the minimum ofG is achieved only at this point, and the value of the minimum is

Km=d+ 2c+3b +2a

². (3.4)

IfK > Km, thenQ⁰(K)=Q(K)∩R⁺_∗²= {(x,y)∈R⁺_∗²|G(x,y)=K}is a nonempty compact component ofQ(K), and through every pointM∈R⁺_∗²passes a unique curveQ⁰(K).

We can thus applyProposition 2.1and obtain the following theorem.

Theorem3.2. Ifa≥0,b≥0,c≥0,d≥0,a+b >0, andb+c+d >0, every solution of the diﬀerence equation (1.2)

un+2un=a+bu_n+1+cu²_n+1

c+dun+1+u²_n+1 (3.5)

is bounded and persistent inR⁺_∗². If (u1,u0)=(,), then(u_n)diverges, the point M_n= (un+1,un)moves on the curveQ⁰(K)which passes throughM0, andK > Km. Moreover the equilibriumLis locally stable.

4. The homographic case

In [5], the authors study the diﬀerence equation u_n+2= αun+1+β

u_nγu_n+1+δ, withα,β,γ,δ≥0,α+β >0,γ+δ >0. (4.1) Ifγ=α=0, we find the classical sequenceu_n+2=(β/δ)/u_nwhich is always 4-periodic.

Ifγ=0,α=0, the sequencevn=(δ/α)unsatisfiesvn+2vn=vn+1+βδ/α²: it is a Lyness sequence, and its behavior is known and given in [3].

So, we supposeγ >0, and thus we can supposeγ=1.

Under this hypothesis, if we suppose that the two quadratic polynomials of (1.2) have a common rootx= −p <0, then (4.1) is a particular case of (1.2). To see this fact, we examine some cases.

(i) Ifδ=0, easy calculation shows that with a=αβ

δ , b=α²

δ +β, c=α, d=α

δ+δ, (4.2)

(1.2) is exactly (4.1) withγ=1.

(ii) If δ=α=0, (4.1) becomesu_n+2u_n=β/u_n+1, which is a classical 3-periodic sequence.

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(iii) Ifδ=0,α >0,β=0,u_n+2=α/u_nis another case of the previous 4-periodic sequence.

(iv) Ifδ=0,α >0,β >0, we putun=β/vn, and obtainvn+2vn=β²/(vn+1+α), which has the form (αvn+1+β)/(vn+1+δ). Thus, (1.2) for (vn) with the valuesa=c= 0,b=β²,d=α, is exactly (4.1) foru_n=β/v_n.

In any cases, (4.1) comes down to (1.2) or to a known sequence (Lyness’ one) or to elementary sequences (3- or 4-periodic). Thus, with the aid of elementary results on Ly- ness’ equation (see [3]), we deduce again fromSection 2the result of [5] about (4.1), but almost without calculation, and we can improve it.

Proposition4.1. The solutions of (4.1) are bounded and persistent, and diverge if(u1,u0) is diﬀerent than the fixed point. Moreover the equilibrium point is locally stable.

Of course, other properties of the solutions of (4.1) will follow from the general property of solutions of (1.2) that we will prove in the following parts, see corollaries of Theorems5.1and7.1, where examples of (4.1) which have IPCB are given.

5. The casea=0

In this part, we solve the case whena=0, which is simple, because an easy birational map reduces the associated quartic curves to cubic ones which give a previous case already solved (see [4]). So we obtain the following general result.

Theorem5.1. Let the diﬀerence equation inR⁺_∗be un+2un= bu_n+1+cu²_n+1

c+dun+1+u²_n+1 withb >0,c≥0,d≥0,c+d >0, (5.1) whose solutions diverge if(u1,u0)=(,). LetL=(,)be the equilibrium, withpositive solution of the equationY³+dY²₋b₌0. LetF(x,y)₌((bx+cx²)/ y(c+dx+x²),x)be the homeomorphism ofR⁺_∗² associated to (5.1):Mn:=(un+1,un)=Fⁿ(M0). LetQb,c,d(K) be the quartic curve with equation

x²y²+dxy(x+y) +cx²+y²+b(x+y)−Kxy=0 (5.2) which passes throughM0=(u1,u0), andQ⁰_b,c,d(K)its positive component, globally invariant under the action ofF.

(a)There exists a well-defined numberθ_b,c,d(K)∈]0, 1/2[such that the restriction ofF toQ⁰_b,c,d(K)is conjugated to a rotation on the circle, of angle2πθb,c,d(K)∈]0,π[.

(b)For everyb,c,dsatisfying the conditions of (5.1) andb²=c³orbd=2c², the diﬀer- ence equation (5.1) has IPCB.

(c)Every integern≥4is the minimal period of some solution of (5.1) for someb,c,d, and some initial pointM0.

One hasb²=c³andbd=2c²if and only if every solution of (5.1) is5-periodic.

Proof. Ifa=0, then the quartic curve (1.6) reduces to (5.2), and then it passes through (0, 0).

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(1) Casec=0 andd >0.

We define the birational map X=

b d 1

x, Y= b

d 1

y, (5.3)

that is the transformation on the solutions (un) of diﬀerence equation (5.1) by the formula

vn= b

d 1

un. (5.4)

Under map (5.3) the quartic (5.2) becomes the cubic of paper [4]:

Γα(K) withα₌

b

d³,K₌_√K

bd, (5.5)

associated to the diﬀerence equation

vn+2vn+1vn=α+vn+1 (5.6)

whose solutions are studied in [4]. Then results ofTheorem 5.1are nothing else but [4, Proposition 8 and Theorem 4].

(2) Casec >0.

We define now the birational map X=b

c 1

x, Y=b c 1

y, (5.7)

that is the transformation on the solutions (un) of diﬀerence equation (5.1) by the formula

vn=b c

1

u_n. (5.8)

Under map (5.7) the quartic (5.3) becomes the cubic of (see [4]) Γα,β(K) withα=b²

c³,β=bd

c²,K=K

c, (5.9)

associated to the diﬀerence equation

vn+2vn=α+βvn+1+v²_n+1

vn+1+ 1 (5.10)

whose solutions are studied in [4]. Then the results inTheorem 5.1are nothing but [4, Proposition 11 and Theorem 6]. The case of 5-periodicity in [4] corresponds to the values α=1 andβ=2 (see [4, Lemma 8]), which gives the end of assertion (d) of the theorem.

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Point (3) of [4, Theorem 6] has to be modified, because here we have onlyα >0 and β≥0 (arbitrary), but in [4] we haveα≥0 andβ >−2^√α. So, in [4, Lemma 10], we must replace the domainDbyD=R⁺²and the functionf()=(1/π) cos⁻¹(1/2)(1−1/^√+ 1) by the functionf()=(1/π) cos⁻¹(1/2)1 + 3/(+ 1). Then it is easy to show that we have only

o

ψ(D) =]0, 1/3[. Thus every integern≥4 is actually a period.

Corollary 5.2. The solutions of (4.1) studied in [5], where αβγ >0, δ =0, satisfy Theorem 5.1.

Remark 5.3. (1) Ifc >0, then solutions (u_n) of (5.1) are rational if and only if thev_n’s are rational, whenb,c,dare rational. Then, in this case, a rational periodic solution of (5.1) may have only periods which belong to the set{3, 4, 5, 6, 7, 8, 9, 10, 12}(see [4]).

But ifc=0, the map (5.3) does not preserve rationality of real numbers, except if b/d=q²withq∈Q⁺_∗. In this case, and withbrational, the periodic rational solutions of (5.1) may have only periods 7 or 10 (see [4, corollary of Proposition 7]).

(2) The 5-periodic caseb²=c³andbd=2c²corresponds to initial Lyness’ sequence:

v_n=√

c/u_nsatisfiesv_n+2v_n=1 +v_n+1.

We give now two easy cases witha=0, which are not covered byTheorem 5.1.

First, the casea=b=0 is given byLemma 3.1: the sequence tends to 0.

Second, we have the following classical result.

Lemma5.4 (case a=c=d=0,b >0). The positive solutions of the diﬀerence equation u_n+2u_n+1u_n=bare3-periodic.

6. General results in the casea >0

It is easy to see that ifa >0 we can suppose thata=1 (putu_n=v_n^√⁴a). We make this hypothesis from now on.

6.1. Points on the diagonal and the birational transformation of the quartic. We know that the quartic curve has two double points at infinity in vertical and horizontal direction, which are ordinary ifd²−4c=0, the asymptotes being then the linesx=r1, x=r2, y=r1, and y=r2, where the r_i are the roots (real or complex) of the equations²+ds+c=0. Moreover, if K > Km, the quarticsQ(K) have no singular point in R⁺_∗². Indeed, if the equation ofQ(K) is p(x,y)−Kxy=0, singular points are given by p_x−K y=0, p_y−Kx=0, andp−Kxy=0. These relations givex p_x=pandy p_y=p, and these last relations are the equations whose solutions are the critical points of the functionG(x,y)=p(x,y)/xy. But we have seen thatGhas no critical point inR⁺_∗²except forL=(,), and so the only finite singular point ofQ(K) inR⁺_∗²would beL, but this point is not onQ(K) ifK > Km.

So we can hope that the quartic curveQ(K) is an elliptic one, and thus that it can be transformed in a regular cubic curve by a birational transformation. To make such a transformation, some point ofQ(K) should disappear, and to preserve the symmetry of the curve with respect to the diagonalδ:x=y, we choose this point on this diagonal.

So the fundamental technical result will be the behavior of the points ofQ(K) on the diagonal.

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f1 f0 −λ0−λ f2 f3

Km t K

G(t, t)

Figure 6.1

Lemma6.1. ForK > Km, the coordinates of the intersection points ofQ(K)with the diagonal δare solutions of the equation

t⁴+ 2dt³+ (2c−K)t²+ 2bt+ 1=0. (6.1) These coordinates are real numbers f1,−λ,f2, f3which satisfy

f1<₋ <₋λ <0< λ < f2< < f3 ifd+b >0, f1= −1

λ<−1<−λ <0< λ=f2<1= < f3=1

λ ifd=b=0. (6.2) Moreover, numbers fiandλare continuous functions ofKon]Km, +∞[, whose limits when K→+∞andK→K_mare

Klim→+_∞λ= lim

K→+_∞f2=0, lim

K→+_∞f1= − lim

K→+_∞f3= −∞, (6.3)

Klim→Km

f2= lim

K→Km

f3=, λ0:= lim

K→Km

λ=(d+)−

(d+)²− 1 ²,

Klim→Km

f1:= f0= −(d+)−

(d+)²− 1 ².

(6.4)

Proof. Formula (6.1) is obvious. Leth_K(t)=t⁴+ 2dt³+ (2c−K)t²+ 2bt+ 1=0. By (3.1) and (3.4) we have

h_K()=d³+ 3b+ 2 + (2c−K)²< d³+ 3b+ 2−²

d+3b + 2

²

=0, (6.5) h_K(0)=1, and soh_K has two roots f2and f3which satisfy 0< f2< < f3. We have also hK(−)=hK()−4d³−4b <0, and thus we have two other roots f1 and−λ which satisfyf1<− <−λ <0. At last,hK(λ)=hK(−λ) + 4dλ³+ 4bλ=4dλ³+ 4bλ≥0. Ifb+d >

0, thus we have 0< λ < f2. Ifb=d=0, the roots of h_K are f1,−λ,λ, and−f1, whose product is 1. This gives (6.2).

Then we remark that the equationh(t)=0 is equivalent to the relationG(t,t)=K. But the graph of the functiont→G(t,t) is easy to determine, seeFigure 6.1. It is immediate from this graph that the roots are continuous functions ofK. Their limits whenK→+∞

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are obvious and given by (6.3). If K→K_m, then f2 and f3 tend to, and f1 and −λ have limits f0and−λ0which are the two other roots of equationh_K_m(t)=0, which has already the double root. Thus these two other roots are easy to obtain, they are given by

(6.4).

Now we write the equation ofQ(K) in the form:

X²Y²+dXY(X+Y) +cX²+Y²+b(X+Y) + 1−KXY=0. (6.6) We make the birational transformationφKdefined in aﬃne coordinates, forXY=λ², by

x= X+λ

XY−λ², y= Y+λ

XY−λ², or X=1 +λx

y , Y=1 +λy

x , (6.7)

or, in homogeneous coordinates, by

X=xt+λx², Y=yt+λy², T=xy, (6.8) or

x=T(X+λT), y=T(Y+λT), t=XY−λ²T². (6.9) OnQ(K) this transformationφK is not defined only at the point (−λ,−λ) ifd+b >0, and at the points (−λ,−λ) and (λ,λ) ifb=d=0.

Now we determine the image ofQ(K) underφ_K. We substitute the second formulas of (6.7) in (6.6). PuttingD:=1 +λ(x+y), easy calculation gives for the left hand of the equation in variablesx,ythe product ofDby the following factor

dλ²−cλ+bxy(x+y) +λcx³+y³+ (c+dλ)(x+y)²+ (d+λ)(x+y)

+2λ²−2dλ−2c−Kxy+ 1 (6.10)

(the coeﬃcient ofx²y²isλ⁴−2dλ³+ (2c−K)λ²−2bλ+ 1 which is 0 because the point (−λ,−λ)∈Q(K)).

So we obtain the straight line∆λwith equation 1 +λ(x+y)=0 and the cubic curve Γ(K) with equation

(x+y)λc(x+y)²+α(K)xy+ (c+dλ)(x+y)²+ (d+λ)(x+y)−β(K)xy+ 1=0, (6.11) where

α(K)=dλ²−4cλ+b, β(K)=K+ 2c+ 2dλ−2λ². (6.12) With second formulas of (6.7), one sees that if (x,y)∈∆λ\ {(−1/λ, 0), (0,−1/λ)}, then (X,Y)=(−λ,−λ) which has no image by φ_K. Identification of the images of Q(K)\ {(−λ,−λ)}andQ⁰(K) is given in the following results.

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Lemma6.2. (1)Ifb+d >0, thenXY > λ²onQ⁰(K), and ifb=d=0, thenXY≥λ²on Q⁰(K), with equality only at the point(λ,λ).

(2)Ifb+d >0, then the positive componentΓ⁰(K)of the cubicΓ(K)is compact, andφK

is a homeomorphism ofQ⁰(K)ontoΓ⁰(K). Ifb=d=0, thenΓ⁰(K)is unbounded and has a point at infinity in directionx=y, which is the image byφ_K of the point(λ,λ)ofQ⁰(K), andφKis a homeomorphism ofQ⁰(K)\ {(λ,λ)}onΓ⁰(K).

Proof. (1) We work here inR⁺_∗², and begin in the case whenb+d >0. Suppose that there is a point (X,Y) in the set{G≤K}, lying on the hyperbolaXY =λ². Then we have X+Y≥2λand

0≥X²Y²+dXY(X+Y) +c(X+Y)²−(2c+K)XY+b(X+Y) + 1

≥λ⁴+ 2dλ³+ (2c−K)λ²+ 2bλ+ 1

=h_K(−λ) + 4dλ³+ 4bλ

=4dλ³+ 4bλ >0,

(6.13)

and this is impossible. So the set {G≤K}, which is connected byProposition 2.1, is contained in one of the two connected components ofR⁺_∗²\ {XY =λ²}. But f2> λby Lemma 6.1, and thusQ⁰(K)⊂ {XY > λ²}.

Now ifb=d=0, we choose (X,Y)=(λ,λ). ThenX+Y >2λifXY=λ², and the same calculation gives, on ({G≤K} \ {(λ,λ)})∩ {XY =λ²}, the impossible inequality 0>0.

But this calculation proves also that{G < K} ∩ {XY=λ²} = ∅. So{G≤K}is contained in{XY≥λ²}or in{XY≤λ²}, and we conclude, with the aid of the point (f3,f3), that {G≤K} \ {(λ,λ)} ⊂ {XY > λ²}.

(2) Ifb+d >0, we have XY > λ² on Q⁰(K), and formulas (6.7) show thatφK is a homeomorphism ofQ⁰(K) onto the positive componentΓ⁰(K) ofΓ(K) (note thatΓ(K) does not intersect the axis{y=0} ∩ {x≥0}nor{x=0} ∩ {y≥0}, by formula (6.11)).

So the setΓ⁰(K) is compact inR⁺_∗².

Ifb=d=0, (6.7) shows thatφKis a homeomorphism ofQ⁰(K)\ {(λ,λ)}ontoΓ⁰(K), and soΓ⁰(K) cannot be bounded. Equation (6.11) becomes

λc(x−y)²(x+y) +c(x+y)²+λ(x+y)−βxy+ 1=0, (6.14) and this proves thatΓ(K) has the point at infinity in the direction of the diagonal. More- over, (6.7) shows that when (X,Y)→(λ,λ) onQ⁰(K), then (x,y) tends to infinity in

directionx=yonΓ⁰(K).

6.2. The algebraic-geometric interpretation of the transformed sequence, and the el- liptic nature of the quartic and cubic curves. We begin with the transformation of the sequenceM_n=(u_n+1,u_n) byφ_K: what is the behavior of the pointsm_n=φ_K(M_n)?

Lemma6.3. Let m_nbe the image inΓ⁰(K)of the sequenceM_n=(u_n+1,u_n)onQ⁰(K)by the birational transformation φK. Then the symmetric point ofmn+1 with respect to the diagonalx=ylies onΓ⁰(K)and on the straight line(P,mn), whereP=(−1/λ, 0)∈Γ(K) (seeFigure 6.2).

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Mn Mn+1

Q⁰(K)

α φK

P

∆λ

mn

mn+1

Γ⁰(K)

Figure 6.2

Proof. The symmetry with respect to the diagonal is preserved byφ_K. We look at the images byφKof the vertical linesX=α. By (6.7), they are the straight lines with equations αy=1 +λx, and all these lines pass through the pointP=(−1/λ, 0). So the lemma is

obvious.

Now we will, as in [3,4], translate the property of sequencem_ngiven byLemma 6.3 into the additionm_n+1=m_n+Pfor a group law on the cubic curveΓ(K). To do this, we need the regularity of the curveΓ(K), that is its elliptic nature. With this objective, we make a new transformationκ, independant fromK, and defined it by

κ(x,y)=(U,V), wherex+y= −U,x−y=V. (6.15) SoΓ(K) becomes a new cubic curveΓ(K) with equation

−U

λcU²+α(K)U²−V² 4

+ (c+dλ)U²−(d+λ)U−β(K)U²−V²

4 + 1=0, (6.16) or

V²α(K)U+β(K)−

b+dλ²U³+γ(K)U²−4(d+λ)U+ 4=0, (6.17) where

γ(K)=4c+ 4dλ−β(K)=2λ²+ 2dλ+ 2c−K. (6.18) Now we need the following result.

Lemma6.4. For everyK > Km,

β(K)>4c+ 3d+b

>0. (6.19)

Proof. It is obvious with formulas (3.1), (3.4), and (6.12), and condition (3.3).

On the other hand, one can see that the quantityα(K) may be zero for some value of K, and positive or negative (see proof ofProposition 6.7). But if 4c²< bd(and thusb >0 andd >0), thenα(K)≥(bd−4c²)/d >0. In the general case of condition (3.3) only, we

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set

p(K)=α(K)

β(K). (6.20)

Now we define a new transformationψK, in aﬃne coordinates, by

U= u

1−p(K)u, V= v

1−p(K)u, (6.21)

or

u= U

1 +p(K)U, v= V

1 +p(K)U, (6.22)

or, in homogeneous coordinates, by

U=u, V=v, W=w−p(K)u, (6.23) or

u=U, v=V, w=W+p(K)U. (6.24) We obtain a new cubic curveE(K).

Remark that if for someKone hasp(K)=0, thenψK=Id.

Ifp(K)=0, then the line with equationU= −1/ p(K) is an asymptote of inflexion of Γ(K) which is sent to the line at infinity byψ_K; this line is a tangent of inflexion to the cubicE(K).

The equation ofE(K) is

v²β=u³b+dλ²+pγ+ 4p²(d+λ) + 4p³

−u²12p²+ 8p(d+λ) +γ+u4(d+λ) + 12p−4. (6.25) Of course, ifp=0, we find again the cubic curveΓ(K) itself.

Now we can prove the essential result of this part.

Proposition6.5. (1)IfK > K_m, the cubic curveE(K)is regular: it is an elliptic curve. So there is onE(K)an abelian group law whose unit element is the point at infinity in vertical direction, and whose addition is defined byA+B+C=0if and only if the three pointsA, B,CofE(K)are on the same straight line (and the opposite−Ais the symmetric ofAwith respect to the u-axis).

(2)Letmnbe the images of pointsmnbyψK◦κ, andPthe image ofPby the same map.

Then, for everyn,m_n+1=m_n+P, and m_n=m0+nP, for the addition of the group law on E(K).

Proof. Equation ofE(K) has the formβv²=P3(u), withβ >0 and deg(P3)≤3. So we know (see [1,6]) thatE(K) is regular if and only if

(i) the coeﬃcient ofu³inP3is nonzero (deg(P3)=3);

(ii) the three roots ofP3are distinct.