Invariant Fibrations for some Birational Maps of C^2

In this article we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them.

It can be seen that if f (x 1 , x 2 ) is a birational map, then the sequence of the degrees of F n satisfies a homogeneous linear recurrence with constant coefficients (see [12] for instance). This is governed by the characteristic polynomial X (x) of a certain matrix associated to F.
The other information we get from X (x) is the dynamical degree, δ(F ), which is defined as The logarithm of this quantity has been called the algebraic entropy of F . It is known that the algebraic entropy is an upper bound of the topological entropy, which in turn is a dynamic measure of the complexity of the mapping. For instance, periodic or integrable birational mappings have zero algebraic entropy.
Birational mappings with zero algebraic entropy have been characterized, see [12] and [4]. From its results we know the existence of some fibrations associated to the mapping, which give almost a complete dynamical information of the mapping.
This family is part of a more general family studied in [10] and [11], which in turn is a generalization of the birational mappings studied by Bedford and Kim in [2]. The goal of this paper is to extract, under affine equivalence, all mappings of type (2) having zero algebraic entropy and give the corresponding invariant fibrations associated to them.
The methodology involves the implementation of the blowing-up technique and the extension of the mappings at the Picard group (see Section 2).
In general, given a parametric family of mappings, to decide for which values of the parameters the mappings are periodic, is not an easy problem (see [9], [5], for instance).
When the mapping is a plane birational mapping it is possible to face that problem (see [2]) and it is fascinating to see how these cases arise, and not only the periodic ones, also all the zero entropy cases.
The paper is organized as follows. In Section 2 we give some preliminary results and we explain how we proceed to find the invariant fibrations associated to zero entropy maps.
Section 3 deals with the subfamily α 2 = 0. The main result is Theorem 11. Similarly in Section 4 we consider the subfamily α 2 = 0 getting Theorem 13.

Preliminary results
Rational mappings F : P C 2 → P C 2 have an indeterminacy set I(F ) of points where F is ill-defined as a continuous map. This set is given by: If F is birational then we can also consider the indeterminacy points of its inverse F −1 .
On the other hand, if we consider one irreducible component V of the determinant of the Jacobian of F , it is known (see Proposition 3.3 in [13]) that F (V ) reduces to a point which belongs to I(F −1 ). The set of these curves which are sent to a single point is called the exceptional locus of F and it is denoted by E(F ).
It is known that the dynamical degree depends on the orbits of the indeterminacy points of the inverse of F under the action of F, see [12,14]. Indeed, the key point is whether the iterates of such points coincide with any of the indeterminacy points of F. When it happens, this orbit is finite.
Sometimes some orbit collision appears. The expression orbit collision refers to the following: Let S ∈ E(F ) which collapses at the point A ∈ I(F −1 ) (we will write S A to describe this behaviour). Following the orbit of A, assume that it ends at a point O ∈ I(F ).
Then, being f birational, (see [12]) it existsĀ ∈ I(F −1 ) withS Ā . When it happens it is said that the orbits of A andĀ collides. This is exactly the behaviour that we get in family (1) and what makes the family so interesting. Given the point ((0, 0), [u : v]) ∈ E p (resp. ((x, y), [x : y])) we are going to represent it by [u : v] Ep (resp. by (x, y) ∈ C 2 or by [1 : x : y] ∈ P C 2 if it is convenient). After every blow up we get a new expanded space X and the induced mapF : X → X. And thenF induces a morphism of groups,F * : Pic(X) → Pic(X) just by taking classes of preimages, where Pic(X) is the Picard group of X (see [1,2]). It is proved that after a finite number of blowing-up's we get a mapF which satisfies F n * = F * n . MapsF satisfying this equality are called Algebraically Stable Maps (AS for short), (see [12]). The characteristic polynomial of the matrix ofF * is the one associated to the sequence of degrees d n := degree F n .

Lists of orbits.
We derive our results by using Theorem 1 below, established and proved in [1,2]. The proof of that is based in the same tools explained in the above paragraph. In order to determine the matrix of the extended map in the Picard group, it is necessary to distinguish between different behaviors of the iterates of the map on the indeterminacy points of its inverse.
The theorem is written for a general family G of quadratic maps of the form G = L • J.
As we will see the maps of family (18), when the triangle is non-degenerate, are linearly conjugated to such a maps. Here L is an invertible linear map and J is the involution in P C 2 as follows: We find that the involution J has an indeterminacy locus I = { 0 , 1 , 2 } and a set of , the elements of this set are determined by a i := G(Σ i − I(J)) = L i for i = 0, 1, 2; see [1].
To follow the orbits of the points of I(G −1 ) we need to understand the following definitions and construction of lists of orbits in order to apply the result of Theorem 1.
We assemble the orbit of a point p ∈ P C 2 under the map G as follows. For a point p ∈ E(G) ∪ I(G) we say that the orbit O(p) = {p}. Now consider that there exits a p ∈ P C 2 such that its n th − iterate belongs to E(G) ∪ I(G) for some n, whereas all the other n − 1 iterates of p under G are never in E(G) ∪ I(G). This is to say that for some n the orbit of p reaches an exceptional curve of G or an indeterminacy point of G. We thus define the orbit of p as O(p) = {p, G(p), ..., G n (p)} and we call it a singular orbit. If for some p ∈ P C 2 in turns out that p and all of its iterates under G are never in E(G) ∪ I(G) for all n, we set as O(p) = {p, G(p), G 2 (p)...} and O(p) is non singular orbit. We now make another characterization of these orbits. Consider that a singular orbit reaches an indeterminacy point of G, this is to say that G n (p) ∈ I(G) but its not in E(G). We call such orbits as singular elementary orbits and we refer them as SE-orbits. To apply Theorem 1 we need to organize our SE orbits into lists in the following way.
Two orbits O 1 = {a 1 , ..., j 1 } and O 2 = {a 2 , ..., j 2 } are in the same list if either j 1 = 2 or j 2 = 1, that is, if the ending index of one orbit is the same as the beginning index of the other. We say that a list of orbits Otherwise it is an open list. For instance, are closed lists.
We now define two polynomials T L and S L which we will use to state Theorem 1. Let Here L runs over all the orbit lists.
This theorem enables us to calculate the characteristic polynomial associated to d n . To this end we have to perform the lists of the orbits of the points in I(F −1 ), but for this we have to do the necessary blow-up's to get an AS mapping.
In order to get AS maps we will use the following useful result showed by Fornaess and Sibony in [14] (see also Theorem 1.14) of [12]: The mapF is AS if and only if for every exceptional curve C and all n ≥ 0 ,F n (C) / ∈ I(F ). (3)

Zero entropy
The following result is quiet useful in our work. It is a direct consequence of Theorem 0.2 of [12]. Given a birational map F of P C 2 , letF be its regularized map so that the induced mapF * : Pic(X) → Pic(X) satisfies (F n ) * = (F * ) n . Then Theorem 2. (See [12]) Let F : P C 2 → P C 2 be a birational map,F be its regularized map and let d n = deg(F n ). Then up to bimeromorphic conjugacy, exactly one of the following holds: • The sequence d n grows quadratically,F is an automorphism and f preserves an elliptic fibration.
• The sequence d n grows linearly and f preserves a rational fibration. In this caseF cannot be conjugated to an automorphism.
• The sequence d n is bounded,F is an automorphism and f preserves two generically transverse rational fibrations.
• The sequence d n grows exponentially.
In the first three cases δ(F ) = 1 while in the last one δ(F ) > 1. Furthermore in the first and second, the invariant fibrations are unique.
We recall that f : When the sequence d n is bounded it can happen that it is periodic or not. For mappings which are not periodic, we have the following result (Theorem A of [4]): Theorem 3. (See [4]) Let F : P C 2 → P C 2 be a non-periodic birational map such that the corresponding sequence of degrees is bounded. Then F is conjugate to an automorphism of P C 2 , which restricts to one of the following automorphisms on some open subset isomorphic to C 2 : (1) (x, y) → (α x, β y), where α, β ∈ C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α β j is generated by (k, 0) for some k ∈ Z.

Invariant fibrations
From Theorem 2 we know the existence of rational invariant fibrations depending on the growth of d n . To find them, we consider V (x, y) = P (x,y) Q(x,y) for some polynomials P (x, y), Q(x, y) without common factors. If V is an invariant fibration, then f sends V = k to V = k .
Note that in case (a) in general the functions P and Q are invariant under f as they satisfy the equation P ·Q(f ) = Q·P (f ) (unless that the denominators of P (f ) or Q(f ) are simplified with Q or P respectively). Similarly for case (b) it follows. In case (c) only Q is invariant as it satisfies the relation Q · P (f ) = (ω 1 P + ω 2 Q) · Q(f ).
Hence we always begin finding invariant algebraic curves. To find them, we introduce the following definition. Given a birational map and given a curve C ⊂ P C 2 we define F (C) := F (C \ I(F ) to be the proper transform of C by F. When C ∩ I(F ) = ∅, we have where m O (C) is the algebraic multiplicity of C at O (see (1), pg. 416, [?]).
The approach is the following. Take an arbitrary curve C and impose that deg As we will see our particular mappings, sometimes depend on a number α which is a zero of certain polynomial P . Then all the calculations have to be made in C[α] (P (α)) [x, y], which in fact make them more complicated ( C[α] (P (α)) is the quotient ring C[α] over the ideal generated by the polynomial P (α)).
Taking into account that α 1 , β 1 and γ 2 are not zero, it can be proved that when α 2 = 0, after an affine change of coordinates f (x, y) can be written as We consider the imbedding (x, y) → [1 : x : y] ∈ P C 2 into projective space and consider the induced map F : P C 2 → P C 2 given by: The indeterminacy locus of F is The set of exceptional curves is given as and the set of exceptional curves of F −1 is given as and letF be the induced map after blowing up the point A 0 . Then the following hold: • IfF 2k (A 1 ) = O 1 for all k ∈ N andF p (A 2 ) = O 0 for some p ∈ N then the characteristic polynomial associated with F is given by and -for p = 0, p = 1 the sequence of degrees d n is bounded, -for p = 2 the sequence of degrees d n grows linearly, -for p > 2 the sequence of degrees d n grows exponentially.
then the characteristic polynomial associated with F is given by and the sequence of degrees grows exponentially. Furthermore δ(F ) → δ * as k → ∞.
• IfF 2k (A 1 ) = O 1 andF p (A 2 ) = O 0 for some p, k ∈ N then the characteristic polynomial associated with F is given by and -for p > 2 (1+k) k the sequence of degrees d n grows exponentially.
-for (k, p) ∈ {(2, 3), (1, 4)} the sequence of degrees d n either, it is periodic or it grows quadratically; Then the characteristic polynomial associated with F is given by and the sequence of degrees grows exponentially with δ(F ) = δ * .
The orbit of A 0 is SE. By blowing up A 0 we get the exceptional fibre E 0 and the new space X. The induced mapF : X → X sends the curve We see that A 1 = O 1 and the exceptional curve S 1 A 1 ∈ S 0 . We observe that the collision of orbits discussed in preliminaries is happening here. The orbit of A 1 underF is as follows: After some iterates we can write the expression ofF 2k ( Observe that for some value of k ∈ N it is possible thatF 2k (A 1 ) = O 1 . This happens when the following condition k is satisfied for some k.
For such k ∈ N the orbit of A 1 is SE. By blowing up the points of this orbit we get the new space X 1 and the induced mapF 1 . Then under the action ofF 1 we have Now if the orbit of A 1 is SE and ifF p 1 (A 2 ) = O 0 that is the orbit of A 2 is also SE for some p ∈ N then we have three SE orbits. If condition k is not satisfied then with the extended mapF we have I(F ) = {O 0 , O 1 }. Therefore we have two options: then p would be greater than zero and since S 0 = T 2 , it would imply that which is not the case (recall that the only points in T 2 which have a preimage are A 1 and A 2 ).
The second case is not possible as A 2 is a fixed point. In the first casẽ Hence the orbit of A 1 must be SE and that condition k must be satisfied for k = r which is a contradiction. It implies that the only available possibility for O 2 to be SE is to have After the blow up process we get The extended mapF 2 is an automorphism when we have three SE orbits.
The above discussion gives us three different cases. 2 , which is given by the greatest root of the polynomial X(x) = x 2 − x − 1. Therefore it has exponential growth.

• Two SE orbits (a): It is the case when
for all k ∈ N. By organizing the orbits into lists we have one closed list For p = 0 and p = 1 the sequence of degrees satisfies d n+3 = d n and d n+4 = d n+3 respectively which corresponds towards boundedness of d n .
For p = 2 we get the polynomial X 2 = x 2 (x + 1)(x − 1) 2 . Looking at the first degrees we get that the sequence of degrees is d n = −1 + 2 n.
Hence X p always has a root λ > 1 and the result follows.
all p ∈ N then there is one open and one closed list and X k = x 2k+1 (x 2 − x − 1) + 1.
We observe that for all the values of k ∈ N , k ≥ 1 the polynomial X k has always a root λ > 1. Therefore f has exponential growth.
• Three SE orbits: In this case we have We have two closed lists as follows: The mapF 2 is an automorphism for all the values (k, p). According to Diller and Favre in [12] the degree growth of iterates of an automorphism could be bounded, quadratic or exponential but it cannot be linear as in such a case the map is never an automorphism. For this we observe the behavior of X (k,p) around x = 1.
The argument for the proof of other values of (k, p) ∈ A (k,p) follows accordingly.
From the above theorem we see that zero entropy cases only appear whenF p ( We are going to study the dynamics of each case separately. Recall that condition k is given by The following proposition considers the case when p = 0. From the above theorem we know that if condition k is not satisfied the sequence d n is bounded and when it is satisfied, d n is a periodic sequence of period 2k + 2. In any case we have to find two generically transverse fibrations. In the second case we present two first integrals functionally independent. We also prove that when d n is periodic, the mapping f (x, y) is itself periodic. y) can be written as and the following hold: • If α 1 = 1 then f (x, y) preserves the two generically transverse fibrations If α k+1 1 = 1 then f is a (2k + 2)−periodic map. In this case W 1 (x, y) and W 2 (x, y) are two independent first integrals, where W i (x, y) := (V i (x, y)) 2k+2 .
• If α 1 = 1 then f (x, y) = 1 + x + y, x 1+y and it preserves the two generically transverse fibrations we know that f has two invariant fibrations. To find them follow the procedure explained in subsection 2.4. We consider an arbitrary cubic projective curve: and we force that C is zero over the indeterminacy points of F, that is, is as follows: The curveC is a degree three algebraic curve. We now impose that C[x 0 : x 1 : , then after some calculations we found (in affine coordinates) The curves Q 1 and Q 2 are invariant algebraic curves while L is an exceptional curve. Taking Now considering the mapping ϕ(x, y) := (V 1 (x, y), V 2 (x, y)), we see that it is a birational mapping and it has the property that ( When α 1 = 1 we see that V 1 or V 2 is a constant function and that it is the unique value of the parameters which has this behaviour. If we take √ 1 = 1 we get the invariant with V 1 (f (x, y)) = −V 1 (x, y). To find V 2 we consider a rational function of type V (x, y) = k 0 +k 1 x+k 2 y+k 3 y 2 1+y where k i ∈ C for i ∈ {0, 1, 2, 3} and imposing V (f (x, y)) = V (x, y) + 1 after some calculations we find V 2 (x, y). Also in this case f (x, y) is birationally conjugated to (−x, y + 1), see Theorem 3 again.
To deal with the case p = 1, that is F (A 2 ) = O 0 , we notice that this condition is equivalent to It is easy to see that it is true if and only if We note that for these maps condition (8) reads as 1 + α 1 + α 2 1 + · · · + α k+1 1 = 0, which implies that α k+2 1 = 1.
Proposition 7. Assume that F (F (A 2 )) = O 0 . Then f (x, y) can be written as and it preserves the fibration with V (f (x, y)) = − 1 ω V (x, y). If ω 4k+6 = 1 for all k ∈ N this fibration is unique. If ω m = (−1) m for some m ∈ N, then f (x, y) is integrable being W (x, y) = V (x, y) m a first integral.
Proof. Now we assume that F 2 (A 2 ) = O 0 . It is easy to see that it is equivalent toF 2 (A 2 ) = O 0 . For the simplification of calculations we consider α 1 = ω 2 . It implies that the coefficients have to satisfy: Taking into account some resultants of E 1 and E 2 we find that the condition F (F (A 2 )) = O 0 gives the maps which appears in (a). When ω 2 − ω + 1 = 0, that is, when α 2 1 + α 1 + 1 = 0 we get the mappings (b).
We note that for the parametric family (a) condition (8) is which implies that ω is a (4 k + 6)−root of unity, while for the two mappings (b), condition k never is satisfied.
Consider f (x, y) that satisfies (a). By looking for invariant curves we find that V (x, y) can be written as shown in statement of (a). A calculation shows that V (f (x, y)) = If 2k+2 i=0 (−1) i ω i = 0 for a certain k ∈ N then we know that the sequence of degrees is periodic of period 4k + 6. We are going to prove that, the map itself is periodic of period 4k + 6. Since d 4k+6 = d 0 = 1, the mapping F 4k+6 is linear, that is: for some constants r i , p i , q i ∈ R. As S 0 is invariant under the action F 2 , it is invariant under the action of F 4k+6 as well. This implies that we can write f 4k+6 (x, y) = (p 0 + p 1 x + p 2 y, q 0 + q 1 x + q 2 y), for some p 0 , p 1 , p 2 , q 0 , q 1 , q 2 ∈ N.
We find that the following two are the fixed points of f and the third one is fixed by f 2 .
Now these points must also be fixed by f 4k+6 . Then by finding the images of f ix 1 , f ix 2 and f ix 3 under the action of f 4k+6 using (11) Also as the sequence of degrees is periodic of period 4 k + 6 this implies that (F * 1 ) 4k+6 fixes the elements in the basis of Picard group. This implies that (F * 1 ) 4k+6 also fixes E 1 that is the blown up fibre at A 2 . Then F 4k+6 fixes the base point A 2 in P C 2 . By utilizing this information and then solving this system of four equations for the values of p 0 , p 1 , p 2 , q 0 , q 1 , q 2 we find that (p 0 , p 1 , p 2 , q 0 , q 1 , q 2 ) = (0, 1, 0, 0, 0, 1) which shows that f 4k+6 (x, y) = (x, y).
Proof. To find the foliations we began looking for degree 3 invariant curves. We only found is the homogeneous polynomial of degree two with C h [1 : x 1 : x 2 ] = C(x 1 , x 2 ). Then we were looking for degree six invariant curves, with the condition that they passes trough the three indeterminacy points O 1 , O 2 and O 3 with multiplicity two.Consequently, its image has also degree six. Forcing that this image coincides with the curve itself we found some of them. For instance, the two numerators of V 1 (x, y) and V 2 (x, y). A computation gives that V 1 (f (x, y)) = α 3 1 V 1 (x, y), V 2 (f (x, y)) = α 2 1 V 2 (x, y) and that they are generically transverse. Clearly W 1 (x, y) and W 2 (x, y) are first integrals of f (x, y) because α 18 1 = 1. From Theorem 4 we know that the sequence of degrees is periodic of periodic 18. To prove that the map is periodic we apply the result of [7], which says that if a map has two independent first integrals, then it is a periodic map.
Proposition 9. Assume that F 3 (A 2 ) = O 0 and that condition k is satisfied for k = 2.
But we claim that the mapping f itself is not periodic. If it were the case, then f k (x, y) = (x, y) for some k multiple of 30. We observe that f sends: Then f 3 (ϕ(y), y) = (ϕ(h(y)), h(y)) where h(y) = u(y) v(y) with If f where a periodic mapping, h also would be periodic. But h has the fixed point y = 1 + α 2 1 and the derivative of h(y) at this points gives zero. And it is a contradiction because periodic maps have the eigenvalues of modulus one at the fixed points.
To prove (b) we begin by proving that the sequence of degrees grows quadratically.
Then the prescribed fibration will be unique. The characteristic polynomial associated to d n is (x + 1) x 2 + x + 1 x 4 + x 3 + x 2 + x + 1 (x − 1) 4 which implies that either, d n grows quadratically or it is periodic. It only depends on the initial conditions, that is on the values of d n for n = 1, 2, . . . , 11. For that mapping we have been able to calculate these numbers: 2, 3,5,8,12,16,22,28,35,43, 52 which implies that that is, d n grows quadratically.
To find V (x, y) we searched for invariant curves and we found one of degree two: −4y 2 + 4x + 1 and one of degree five, the numerator of V (x, y). Taking the quotient of them, we verified that it satisfies V (f (x, y)) = V (x, y).
The last class with zero entropy is when p = 4 with k = 1. The condition k = 1 says From the proof and notations of Theorem 4 we know that: Hence, if A 2 ∈ S 1 , i.e., Then either: (a) The map f (x, y) can be written as and it preserves the unique elliptic fibration can be written as f (x, y) = x + y, x y − 1 (15) and it preserves the unique elliptic fibration with V (f (x, y)) = −V (x, y). Furthermore f is integrable, being W (x, y) = V (x, y) 2 a first integral of f.
(c) The map f (x, y) can be written as f (x, y) = α 1 x + y, x y + 1 with α 2 1 + 1 = 0 (16) and it preserves the unique elliptic fibration y) can be written as and it preserves the unic elliptic fibration To see the uniqueness of the fibrations we have to prove that d n grows quadratically.
The characteristic polynomial associated to d n is (x − 1) 4 (x + 1) 2 (x 2 + 1)(x 2 + x + 1) which implies that either, d n grows quadratically or it is periodic. It only depends on the initial conditions, that is on the values of d n for n = 1, 2, . . . , 10. For each one of the mappings which appear in the statement, we have been able to calculate these numbers. In the four cases they give 2, 3, 5, 7, 11, 15, 20, 25, 32, 39, which implies that In order to prove (a) we find the family of invariant curves λ (α 0 xy − x 2 y + xy 2 ) + µ (y − 1) = 0. Then taking V = P Q with and P = α 0 xy − x 2 y + xy 2 and Q = y − 1 we have that V (f (x, y)) = V (x, y).
To prove (b) we easily see that and hence x y (y + x) is an invariant cubic. Then taking V as the quotient of a conic and the invariant cubic and imposing V (f (x, y)) = k V (x, y) we found that the conic can be taken as −2 y 2 + 2 x + y + 1 and k = −1.
To prove (c) we find that the straight line α 1 y +α 1 −2 x−y −1 = 0 is sent to the straight line −1 + α 1 − 2 y = 0 and viceversa, which implies that their product is an invariant curve of degree two. Also it can be seen that and hence x y (α 1 y − x) is an invariant cubic. Taking V as the quotient of this invariant curves we get that V (f (x, y)) = α 1 V (x, y) and the result follows To see (d) we began searching invariant curves of degree three and we found (in projective coordinates) Now we take a conic that passes through two indeterminacy points of F , and we impose that its image (a conic again) also passes through two indeterminacy points of F. This gives a third conic which we impose to be equal to the first one. With this we find Q 1 , Q 2 , Q 3 .
Then Q 1 · Q 2 · Q 3 is an invariant curve of degree six. Taking V as the quotient of Q 1 · Q 2 · Q 3 over C[1 : x : y] 2 we get that V (f (x, y)) = α 2 1 V (x, y).
We consider the induced map in the projective plane : F : P C 2 → P C 2 given by The indeterminacy sets of Furthermore the exceptional curves of F and F −1 are the following: Theorem 12. Let f (x, y) be a map of type (18) with γ 2 = 0, α 1 = 0, β 1 = 0 and suppose that α 2 = 0. If f p (α 0 , β 2 ) = (0, 0) for some p ∈ N then the characteristic polynomial associated with f is given by and the sequence of degrees of f is periodic with period 2p + 2. If no such p exists then the characteristic polynomial associated with f is and the sequence of degrees d n grows linearly.
Proof. Observe that S 0 A 0 = O 2 and S 1 A 1 = O 1 . Hence we blow up the points A 0 , A 1 getting the exceptional fibres E 0 , E 1 . Let X be the new space and letF : X → X be the corresponding map on X. Then the mapF sends the curve S 0 → E 0 → S 0 and We observe that no new indeterminacy points are created therefore I(F ) = Assume that there exists p ∈ N such thatF p (A 2 ) = O 0 . Then we blow up A 2 ,F (A 2 ),  : HenceF 1 : X 1 → X 1 is an AS map and also an automorphism. Taking into account that Then by using Theorem 1 we find that the characteristic polynomial associated to F is X = (x p+1 + 1)(x − 1) 2 (x + 1).
As our mapF 1 is an automorphism then by using the results from Diller and Favre in [12] we see that also cannot have linear growth. Therefore we must have c 1 = c 3 = 0. Hence the sequence of degrees must be periodic. This implies that d 2p+2+n = d n i.e. the sequence of degrees is periodic with period 2p + 2. If p is odd then d n is also periodic of period 2p + 2. Then δ(F ) is determined by the polynomial (x − 1) 2 (x + 1), and δ(f ) = 1. The sequence of degrees is d n = 5 4 + 1 2 n − 1 4 (−1) n .
Proof. If f p (α 0 , β 2 ) = (0, 0) for all p ∈ N then from the above theorem we know that d n grows linearly, and hence we know that f (x, y) has a unique invariant fibration. Clearly V 1 (x, y) = x is an invariant fibration and when α 1 = 1 and α 0 = 0, V 1 (x, y) is a first integral.
When α n 1 = 1 the function h(x) is periodic of period n and hence W (x, y) is a first integral of f (x, y).
Now assume that f p (α 0 , β 2 ) = (0, 0) for a certain p ∈ N. From Theorem (12) we know that the sequence of degrees d n is 2p + 2 periodic. We are going to see that f (x, y) itself is a periodic map. Since the map F 2p+2 is linear, we can consider that for some constants r i , p i , q i ∈ R the map F 2p+2 can be written in the following form: F 2p+2 [x 0 : x 1 : x 2 ] = [r 0 x 0 + r 1 x 1 + r 2 x 2 : p 0 x 0 + p 1 x 1 + p 2 x 2 : q 0 x 0 + q 1 x 1 + q 2 x 2 ].