± bi
εdi jdki,∓ ai
εdi jdki
. (49)
The singular orbits of the map are as follows:
L(−1)−→B−(1)−→B+(1)−→L(+1),
L(−2) −→B−(2) −→B+(2)−→L(+2), (50) whereL(∓i)denotes the line through the points B±(j), B±(k).
Proof The singular orbits are a consequence of Proposition3.2and Theorem3.3. It can be verified by straightforward computations that the points (49) are base points of
the pencil of invariant curvesEλ.
With (25) we see that the point B−(3)is a fixed point ofφ+while B+(3)is a fixed point ofφ−. Therefore, they participate in patterns
L(−3)−→B−(3), B+(3)−→L(+3),
which do not qualify as singularity confinement patterns [15,21] and need not be blown up.
7.1 Lifting the map to an analytically stable map
We blow up the planeCP2 at the four base points B−(i),B+(i),i = 1,2, and denote the corresponding exceptional divisors byEi,0,Ei,1,i =1,2. The resulting blow-up surface is denoted byX. On this surfaceφ+is lifted to an analytically stable mapφ+
acting on the exceptional divisors according to the scheme [compare with (50)]
L(−1)−→E1,0−→E1,1−→L(+1), L(−2)−→E2,0−→E2,1−→L(+2),
whereL(±i)denotes the proper transform of the lineL(±i).
We compute the induced pullback map on the Picard groupφ∗+:Pic(X)→Pic(X).
LetH∈Pic(X)be the pullback of the class of a generic line inCP2. LetEi,n∈Pic(X), fori =1,2 andn =0,1, be the class ofEi,n. Then the Picard group is
Pic(X)=ZH 2
i=1
1 n=0
ZEi,n.
Fig. 4 The curvesE0,E∞,E0.1in resp. red, blue and green for1(x,y)= x+y,2(x,y) =x−y, 3(x,y)=xandε=1 (Color figure online)
The rank of the Picard group is 5. The induced pullbackφ+∗: Pic(X)→ Pic(X)is determined by (3).
With Theorem2.3we arrive at the system of recurrence relations for the degree d(m):
⎧⎨
⎩
d(m+1)=2d(m)−μ1(m)−μ2(m), μ1(m+2)=d(m)−μ2(m),
μ2(m+2)=d(m)−μ1(m),
with initial conditions d(0) = 1,μ1(m) = 0, form = 0,1, andμ2(m) = 0, for m=0,1. The solution to this system of recurrence relations is given by
d(m)=2m,
μi(m)=m−1, i =1,2. (51)
The sequenced(m)grows linearly.
8 The case(1,2,3)=(n,1,−1)
By Theorem3.3this case corresponds to the orbit data(n1,n2)=(1,n),(σ1, σ2)= (1,2). In this case, we consider the Kahan mapφ+: C2 → C2 corresponding to a
quadratic vector field of the form
˙
x= 23(x)
n1−1(x)J∇H(x), H(x)= n1(x)2(x) 3(x) . The casen=1 was studied in [20].
For1(x)=x,2(x)=x+y,3(x)=x−ythe vector field reads x˙=2x2,
˙
y= −nx2+ny2+2x y, and the Kahan discretization (13) reads
x−x=4εx x,
y−y=2ε(−nx x+ny y+x y+xy).
Proposition 8.1 The Kahan mapφ+:C2→C2admits an integral of motion H(x) = H(x)
P(x), where
P(x)=
k∈I
(εd23k1(x)+1)(εd23k1(x)−1),
for I = {1,3,5, . . . ,n−1}if n is even and I = {2,4,6, . . . ,n−1}if n is odd.
Proof Note that the following identity holds:
−d123(x)−d312(x)=d231(x). (52) Then, using (52), from Eq. (14) it follows that
1(x)= 1(x)
2εd231(x)+1. (53)
Moreover, multiplying (15) by3(x)and (16) by2(x)and then subtracting the second equation from the first equation and again applying (52), we arrive at
2(x)
3(x) = −2(x)(εd23(n+1)1(x)+1) 3(x)(εd23(n−1)1(x)−1). On the other hand, from (53) it follows that
εd23k1(x)±1=εd23(k±2)1(x)±1 2εd231(x)+1 ,
and therefore, withhk±(x)=(εd23k1(x)±1), we find P(x)
P(x)= h−−1(x)h1−(x)· · ·hn−−3(x)·h3+(x)h5+(x)· · ·hn++1(x) (h2+(x))n·h1−(x)h3−(x)· · ·hn−−1(x)·h1+(x)h3+(x)· · ·hn+−1(x)
= − hn++1(x) (h2+(x))nhn−−1(x), ifnis even, and
P(x)
P(x)= h0−(x)h2−(x)· · ·hn−−3(x)·h4+(x)h6+(x)· · ·hn++1(x) (h2+(x))n−1·h2−(x)h4−(x)· · ·hn−−1(x)·h2+(x)h4+(x)· · ·hn+−1(x)
= − hn++1(x) (h2+(x))nhn−−1(x),
ifnis odd. This proves the claim.
With Theorem2.3we arrive at the system of recurrence relations for the degreed(m):
⎧⎨
⎩
d(m+1)=2d(m)−μ1(m)−μ2(m), μ1(m+1)=d(m)−μ2(m),
μ2(m+n)=d(m)−μ1(m),
with initial conditionsd(0)=1,μ1(0)=0 andμ2(m)=0, form=0, . . . ,n−1.
The generating functions of the solution to this system of recurrence relations are given by
d(z)=1+2z+ · · · +nzn−1+(n+1)zn
1−z , (54)
μ1(z)=z+2z2+ · · · +(n−1)zn−1+ nzn 1−z, μ2(z)= zn
1−z.
Note that the degrees ofφ+k grow linearly fork=1, . . . ,n−1 and stabilize ton+1 forkn. This seems to be the first example of a birational map of deg=2 with such behavior.
Acknowledgements The author would like to thank Matteo Petrera and Yuri Suris for their critical feedback on this manuscript.
Funding Open Access funding enabled and organized by Projekt DEAL.
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