This is a preprint of: “Dynamical Classification of a Family of Birational Maps of C 2 via Algebraic Entropy”, Sundus Zafar, Anna Cima, Qual. Theory Dyn. Syst., vol. 18(2), 631–652, 2019. DOI: [10.1007/s12346-018-0304-1] Dynamical Classification of a Family of Birational Maps of C2 via Algebraic Entropy∗ Anna Cima(1) and Sundus Zafar(1) (1) Dept. de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain cima@mat.uab.cat, sundus@mat.uab.cat Abstract This work dynamically classifies a 9−parametric family of birational maps f : C2 → C . From the sequence of the degrees dn of the iterates of f, we find the dynamical 2 degree δ(f ) of f . We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim in [4] as one of its subfamilies. Mathematics Subject Classification 2010: 14E05, 26C15, 34K19, 37B40, 37C15, 39A23, 39A45. Keywords: Birational maps, Algebraic entropy, First Integrals, Fibrations, Blowing-up, Integrability, Periodicity, Chaos. 1 Introduction In this work we consider the family of fractional maps f : C2 → C2 of the form:   β0 + β1 x + β2 y f (x, y) = α0 + α1 x + α2 y, , (1) γ0 + γ1 x + γ2 y where the parameters are complex numbers. This family of maps can be extended to the projective plane P C2 by considering the embedding (x1 , x2 ) ∈ C2 7→ [1 : x1 : x2 ] ∈ P C2 into projective space. The induced map ∗ Acknowledgements. The first author is supported by Ministry of Economy, Industry and Competi- tiveness of the Spanish Government through grants MINECO/FEDER MTM2016-77278 and also supported by the grant 2014-SGR-568 from AGAUR, Generalitat de Catalunya. The GSD-UAB Group is supported by the Government of Catalonia through the SGR program. It is also supported by MCYT through the grant MTM2008 − 03437. http://www.gsd.uab.cat 1 F : P C2 → P C2 has three components Fi [x0 : x1 : x2 ] , i = 1, 2, 3 which are homogeneous polynomials of degree two. For general values of the parameters the three components don’t have a common factor: we say that these maps have degree two. Similarly we can define the degree of F n = F ◦ · · · ◦ F for each n ∈ N. It can be seen that if f (x1 , x2 ) is a birational map, then the sequence of its degrees satisfies a homogeneous linear recurrence with constant coefficients (see [18] for instance or Section 3). 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 it’s largest real root, and is defined as 1 δ(F ) := lim (deg(F n )) n , (2) n→∞ see [3, 4, 5, 6, 17, 18]. The logarithm of this quantity has been called the algebraic entropy. The application of algebraic entropy in the field of dynamical systems has been growing in recent years, see for instance [3, 4, 5, 6, 7, 8, 11, 17, 18]. On the other hand, the study of the dynamics generated by birational mappings in the plane is also a current issue, see for instance [1, 3, 4, 5, 17] also [14, 15, 16, 20, 21, 22, 23, 25, 24, 28]. It is known (see [26]) 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. The algebraic entropy for the maps (1) highly depends on the choice of parameters. For this reason our study includes all the possible values of the parameters of f to determine the growth rate dn and δ(F ). Therefore the results that we get can be seen as a dynamical classification of family (1). Furthermore, they generalize the results obtained in [4], which the authors consider the subfamily of (1)with α0 = 0, α1 = 0 and α2 = 1. Birational mappings F : P C2 → P C2 have an indeterminacy set I(F ) of points where F is ill-defined as a continuous map. This set is given by: I(F ) = {[x0 : x1 : x2 ] ∈ P C2 : F1 [x0 : x1 : x2 ] = 0, F2 [x0 : x1 : x2 ] = 0, F3 [x0 : x1 : x2 ] = 0]}. 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 [17]) that F (V ) reduces to a point in 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 [18, 19, 27]. Indeed, the key point is whether the iterates of such points coincide with any of the indeterminacy points of F. Generically our map F has three indeterminacy points. The exceptional locus is formed by three straight lines, each two of them intersecting on a single indeterminate point of F . We call them non degenerate mappings. But there is a subfamily such that the exceptional 2 locus is formed by only two straight lines. We call these mappings degenerate mappings and they are studied in the paper [10]. Hence we are not going to consider them here. To find δ(F ) and dn and to study its behaviour, we use a Theorem of Bedford and Kim (see [3]) to find the characteristic polynomial which provides dn . The results obtained in this paper are the starting point for studying the dynamic properties of the elements of family (1). We can expect certain types of behaviors, for instance, if we want to know the mappings which are globally periodic, we have to look at the ones whose sequence of degrees is periodic. To find the mappings that are integrable (i. e., mappings that preserve the level curves of some rational function) or mappings that preserve some fibrations, we have to look for the mappings whose sequence of degrees dn grows linearly or quadratically in n, see [18]. We can encounter chaos whenever dn grows exponentially. As a continuation of this work, in the following articles, ”Zero entropy for some birational maps of C2 ”, see [9], and ”Finding invariant fibrations for some birational maps of C2 ”, see [10], we give all the maps of type (1) which have zero entropy in the particular cases γ1 = 0 in the first one and for the degenerate cases in the second one, giving explicitly the invariants when they exist. The details of prerequisites and results of this work can be found in [12]. The article is organized as follows: The main results are announced in Section 2. The preliminary results which include the basic settings of the work, some background of bira- tional maps and Picard group and the structure of the orbits’lists are introduced in Section 3. In Section 4 we give the proof of the results. Finally in Section 5 we present the rest of the zero entropy mappings which are not included in the two mentioned papers [9] and [10]. 2 Main results The results that we find are presented in the following theorems 1, 2, 3, 4 and 5. In all of them we consider that the coefficients of the map f are such that f is a birational map and F has degree two (see Lemma 5). Consider the family of fractional maps f : C2 → C2 :   β0 + β1 x + β2 y f (x, y) = α0 + α1 x + α2 y, , γ0 + γ1 x + γ2 y where the parameters are complex numbers. We call F [x0 : x1 : x2 ] = [F1 [x0 : x1 : x2 ] : F2 [x0 : x1 : x2 ] : F3 [x0 : x1 : x2 ]], (3) 3 the extension of f (x, y) to the projective plane, where F1 [x0 : x1 : x2 ] = x0 (γ0 x0 + γ1 x1 + γ2 x2 ), F2 [x0 : x1 : x2 ] = (α0 x0 + α1 x1 + α2 x2 )(γ0 x0 + γ1 x1 + γ2 x2 ), F3 [x0 : x1 : x2 ] = x0 (β0 x0 + β1 x1 + β2 x2 ). The indeterminacy set of F [x0 : x1 : x2 ] is I(F ) = {O0 , O1 , O2 }, with O0 = [(βγ)12 : (βγ)20 : (βγ)01 ], O1 = [0 : α2 : −α1 ], O2 = [0 : γ2 : −γ1 ], (4) where (βγ)ij := βi γj − βj γi for i, j ∈ {0, 1, 2}. By calling f −1 (x, y) the inverse of f (x, y) and by F −1 [x0 : x1 : x2 ] its extension on P C2 , also a indeterminacy set I(F −1 ) exists: I(F −1 ) = {A1 , A2 , A3 }, with A0 = [0 : 1 : 0] , A1 = [0 : 0 : 1], A2 = [(βγ)12 (αγ)12 : (α0 (βγ)12 − α1 (βγ)02 + α2 (βγ)01 ) (αγ)12 : (αβ)12 (βγ)12 ]. (5) √ 1+ 5 We denote by δ ∗ = 2 the golden mean, which is the largest root of the polynomial x2 − x − 1. Theorem 1. Let F : P C2 → P C2 be a birational degree two non degenerate map of type (1) and suppose that α1 , α2 , γ1 , γ2 are all non zero. Then either, (i) If it exists p ∈ N such that F p (A2 ) = O0 , then the characteristic polynomial associated with F is Wp = xp+2 − 2xp+1 + x − 1, δ(F ) is given by the largest root of the polynomial Wp and dn grows quadratically. (ii) If no such p exists then δ(F ) = 2 and dn grows quadratically. Notice that Theorem 1 says us that family (1) generically has dynamical degree equal 2. Theorem 2. Let F : P C2 → P C2 be a birational degree two non degenerate map of type (1) and suppose that γ1 = 0. Then γ2 , α1 , β1 are non zero and the following hold: 1. Assume that α2 = 0 and let F̃ be the extension of F after blowing-up the points A0 , A1 . If F̃ p (A2 ) = O0 for some p ∈ N then the characteristic polynomial associated with F is given by Xp = (xp+1 + 1)(x − 1)2 (x + 1), 4 and the sequence of degrees dn of F is periodic with period 2p + 2. If no such p exists then the characteristic polynomial associated with F is X = (x − 1)2 (x + 1), and the sequence of degrees dn grows linearly. 2. Assume that α2 6= 0 and let F̃ be the induced map after blowing up the point A0 . Then the following hold: • If F̃ p (A2 ) = O0 for some p ∈ N and F̃ 2k (A1 ) 6= O1 for all k ∈ N then the characteristic polynomial associated with F is given by Xp = xp+1 (x2 − x − 1) + x2 , and – for p = 0, p = 1 the sequence of degrees dn is bounded, – for p = 2 the sequence of degrees dn grows linearly, – for p > 2 the sequence of degrees dn grows exponentially. • Assume that F̃ 2k (A1 ) = O1 for some k ∈ N. Let F̃1 be the induced map after we blow-up the points A0 , A1 , F̃ (A1 ), . . . , F̃ 2k (A1 ) = O1 . If F̃1p (A2 ) 6= O0 for all p ∈ N, then the characteristic polynomial associated with F is given by Xk = x2k+1 (x2 − x − 1) + 1, and the sequence of degrees grows exponentially. Furthermore δ(F ) → δ ∗ as k → ∞. • If F̃ 2k (A1 ) = O1 and F̃1p (A2 ) = O0 for some p, k ∈ N then the characteristic polynomial associated with F is given by X(k,p) = xp+1 (x2k+3 − x2k+2 − x2k+1 + 1) + x2k+3 − x2 − x + 1, and 2 (1+k) – for p > k the sequence of degrees dn grows exponentially for all p, k ∈ N; – for (p, k) ∈ {(3, 2), (4, 1)} the sequence of degrees dn is periodic or grows quadratically; – for (p, k) ∈ {(0, k), (1, k), (2, k), (3, 1)} the sequence of degrees dn is peri- odic. 5 • Assume that F̃ 2k (A1 ) 6= O1 and F̃ p (A2 ) 6= O0 for all k, p ∈ N. Then the charac- teristic polynomial associated with F is given by X (x) = x2 − x − 1, and the sequence of degrees grows exponentially with δ(F ) = δ ∗ . Theorem 3. Let F : P C2 → P C2 be a birational degree two non degenerate map of type (1) and suppose that γ2 = 0. Then γ1 , α2 , β2 are non zero and the following hold: 1. Let F be the map for α1 = 0 and let F̃ be the induced map after blowing up the points A0 , A1 . If F̃ p (A2 ) = O0 for some p ∈ N then the characteristic polynomial associated with F is given by Yp = xp+1 (x3 − x − 1) + (x3 + x2 − 1), and • for p ∈ {0, 1, 2, 3, 4, 5} the sequence of degrees dn is periodic of period 6, 5, 8, 12, 18 and 30 respectively; • for p = 6 the sequence of degrees dn it grows quadratically or it is periodic of period 30; • for p > 6 the sequence of degrees dn grows exponentially. If no such p exists then the characteristic polynomial associated with F is given by Y = x3 − x − 1, and the sequence of degrees grows exponentially with δ(F ) = δ∗ . 2. Let F be the map for α1 6= 0 and let F̃ be the induced map after blowing up the point A1 . Then the following hold: • If F̃ p (A2 ) = O0 for some p ∈ N then the characteristic polynomial associated with F is given by Yp = xp+1 (x2 − x − 1) + x2 − 1, and the sequence of degrees has exponential growth rate. Furthermore δ(F ) → δ ∗ as p → ∞. • If F̃ q (A2 ) = O1 for some q ∈ N then the characteristic polynomial associated with F is given by Yq = xq+1 (x2 − x − 1) + x2 , and for q ≥ 2: 6 – The sequence of the degrees grows linearly when q = 2. – The sequence of the degrees grows exponentially when q > 2. For q ∈ {0 , 1} there are no such mappings. • Assume that F̃ p (A2 ) 6= O0 and F̃ q (A2 ) 6= O1 for all q, p ∈ N. Then the charac- teristic polynomial associated with F is given by Y(x) = x2 − x − 1, and the sequence of degrees grows exponentially with δ(F ) = δ ∗ . Theorem 4. Let F : P C2 → P C2 be a birational degree two non degenerate map of type (1) and suppose that γ1 6= 0, γ2 6= 0 and α1 α2 = 0. Then: 1. Assume that α1 = 0 and let F̃ be the induced map after blowing up the point A0 . Then the following hold: • If F̃ p (A1 ) = O0 for some p ∈ N and F̃ q (A2 ) 6= O0 for all q ∈ N then the characteristic polynomial associated with F is given by Zp = xp+1 (x2 − x − 1) + x2 , and for p ≥ 2: – The sequence of the degrees grows linearly for p = 2. – The sequence of the degrees grows exponentially for p > 2. For p ∈ {0 , 1} there are no such mappings. • If F̃ q (A2 ) = O0 for some q ∈ N and F̃ p (A1 ) 6= O0 for all p ∈ N then the characteristic polynomial associated with F is given by Zq = xq+1 (x2 − x − 1) + x2 − 1, and the sequence of degrees has exponential growth rate. Furthermore δ(F ) → δ ∗ as q → ∞. • If F̃ p (A1 ) = O0 and F̃ q (A2 ) = O0 for some p, q ∈ N, then p 6= q and – for p > q the characteristic polynomial associated with F is Zq . – for p < q the characteristic polynomial associated with F is Zp . • If F̃ p (A1 ) 6= O0 and F̃ q (A2 ) 6= O0 for all p, q ∈ N then the characteristic poly- nomial associated with F is given by Z(x) = x2 − x − 1, and the sequence of degrees grows exponentially with δ(F ) = δ ∗ . 7 2. Assume α2 = 0 and let F̃ be the induced map after blowing up the point A1 . If there exists some p ∈ N such that F̃ p (A2 ) = O0 , then the characteristic polynomial associated with F is given by Zp = (xp+1 + 1)(x − 1)2 , and for all p ∈ N the sequence of degrees dn grows linearly. If no such p exists then the characteristic polynomial associated with F is given by Z = (x − 1)2 , and dn grows linearly. 3 Preliminary results 3.1 Settings Consider   β0 + β1 x + β2 y f (x, y) = α0 + α1 x + α2 y, . γ0 + γ1 x + γ2 y The exceptional locus of F [x0 : x1 : x2 ] is E(F ) = {S0 , S1 , S2 }, where S0 = {x0 = 0}, S1 = {γ0 x0 + γ1 x1 + γ2 x2 = 0}, S2 = {(α1 (βγ)02 − α2 (βγ)01 ) x0 + α1 (βγ)12 x1 + α2 (βγ)12 x2 = 0}, and the exceptional locus of F −1 [x0 : x1 : x2 ] is E(F −1 ) = {T0 , T1 , T2 }, where T0 = {(γ0 (αβ)12 − γ1 (αβ)02 + γ2 (αβ)01 ) x0 − (βγ)12 x1 = 0} , T1 = {(αβ)12 x0 − (αγ)12 x2 = 0}, T2 = {x0 = 0}. It is easy to see that F maps each Si to Ai where the A0i s are defined in (5) and that the inverse of F maps Ti to Oi for i ∈ {0, 1, 2}, see (4). To specify this behaviour we write F : Si  Ai (also F −1 : Ti  Oi ). We are interested in the mappings (1) when they are birational maps which are not degree one maps. Next lemma informs about the set of parameters which are available in this study. Also the degenerate case and the non degenerate cases are distinguished. We recognize the non degenerate case when F has three distinct exceptional curves. When f has two exceptional curves of such type, then we are in degenerate case. Recall that a birational map is a map f : C2 → C2 with rational components such that there exists an algebraic curve V and another rational map g such that f ◦ g = g ◦ f = id in C2 \ V. 8 Lemma 5. Consider the mappings   β0 + β1 x1 + β2 x2 f (x1 , x2 ) = α0 + α1 x1 + α2 x2 , , (γ1 , γ2 ) 6= (0, 0) 6= (α1 , α2 ). γ0 + γ1 x1 + γ2 x2 Then: (a) The mapping f is birational if and only if the vectors (β0 , β1 , β2 ), (γ0 , γ1 , γ2 ) are linearly independent and ((αβ)12 , (αγ)12 ) 6= (0, 0), ((αγ)12 , (βγ)12 ) 6= (0, 0), and either ((αβ)12 , (βγ)12 ) 6= (0, 0) or (β1 , β2 ) = (0, 0). (b) The mapping f is degenerate if and only if (βγ)12 = 0 or (αγ)12 = 0. Proof. The conditions in (a) are necessary for f to be invertible as if the vectors (β0 , β1 , β2 ), (γ0 , γ1 , γ2 ) are linearly dependent then the second component of f is a constant, also if ((αβ)12 , (αγ)12 ) = (0, 0) or ((αγ)12 , (βγ)12 ) = (0, 0) then f only depends on α1 x1 + α2 x2 or on γ1 x1 + γ2 x2 . If ((αβ)12 , (βγ)12 ) = (0, 0) and (β1 , β2 ) 6= (0, 0) then f only depends on β1 x1 + β2 x2 . Now assume that conditions (a) are satisfied. Then the inverse of f which formally is   −1 −(αβ)02 + β2 x + (αγ)02 y − γ2 xy (αβ)01 − β1 x + (αγ)10 y + γ1 xy f (x, y) = , , (αβ)12 − (αγ)12 y (αβ)12 − (αγ)12 y is well defined. Furthermore the numerators of the determinants of the Jacobian of f and f −1 are α1 (βγ)02 − α2 (βγ)01 + α1 (βγ)12 x + α2 (βγ)12 y (6) and α0 (βγ)12 − α1 (βγ)02 + α2 (βγ)01 − (βγ)12 y, (7) respectively. It is easily seen that conditions (a) imply that both (6) and (7) are not identically zero. Hence, f ◦ f −1 = f −1 ◦ f = id in C2 \ V, where V is the algebraic curve determined by the common zeros of (6) and (7). To see (b) we know that since Si maps to Ai , this implies that the points A0 , A1 , A2 are not all distinct. Since A0 6= A1 we have two possibilities: A0 = A2 or A1 = A2 . Condition A0 = A2 writes as (βγ)12 (αγ)12 = 0 and (αβ)12 (βγ)12 = 0. From (a), the vector ((αβ)12 , (αγ)12 ) 6= (0, 0). Hence (βγ)12 must be zero. In a similar way it is seen that A1 = A2 if and only if (αγ)12 = 0. 3.2 Birational mappings and Picard group Given the birational map f let F [x0 : x1 : x2 ] be the extension of f (x1 , x2 ) at P C2 and consider I(F ) and E(F ). To get rid of indeterminacies we do a series of blowups. More 9 precisely, if F k (Ai ) = Oj we perform the blowingup at the points Ai , F (Ai ), . . . , F k (Ai ) = Oj . Given a point p ∈ C2 , let (X, π), be the blowing-up of C2 at the point p. Then, π −1 p = π −1 (0, 0) = {((0, 0), [u : v])} := Ep ' P C1 and if q = (x, y) 6= (0, 0), then π −1 q = π −1 (x, y) = ((x, y), [x : y]) ∈ X. Given the point ((0, 0), [u : v]) ∈ Ep (resp. ((x, y), [x : y])) we are going to represent it by [u : v]Ep (resp. by (x, y) ∈ C2 or by [1 : x : y] ∈ P C2 if it is convenient). After every blow up we get a new expanded space X and the induced map F̃ : X → X. Hence in this work we deal with complex manifolds X obtained after performing a finite sequence of blow-ups. Indeterminacy sets and exceptional locus can also be defined if we consider meromorphic functions defined on complex manifolds. If X is a complex manifold we are going to consider the Picard group of X, denoted by Pic(X). Then Pic(P C2 ) is generated by the class of L, where L is a generic line in P C2 . As usual, given a curve C on C2 , the strict transform of C is the adherence of π −1 (C \ {p}), in the Zariski topology, and we denote it by Ĉ. If the base points of the blow-ups are {p1 , p2 , . . . , pk } ⊂ P C2 and Ei := π −1 {pi } then it is known that Pic(X) is generated by {L̂, E1 , E2 , . . . , Ek }, where L is a generic line in P C2 (see [3, 4]). Furthermore π : X −→ P C2 induces a morphism of groups π ∗ : Pic(P C2 ) −→ Pic(X), with the property that for any complex curve C ⊂ P C2 , X π ∗ (C) = Ĉ + mi Ei , (8) where mi is the algebraic multiplicity of C at pi . On the other hand, if F is a birational map defined on P C2 , then there is a natural extension of F on X, which we denote by F̃ . And F̃ induces a morphism of groups, F̃ ∗ : Pic(X) → Pic(X) just by taking classes of preimages. The interesting thing here is that k X F̃ ∗ (L̂) = d L̂ + ci Ei , ci ∈ Z i=1 where d is the degree of F. By iterating F, we get the corresponding formula by changing F by F n and d by dn . In order to deduce the behavior of the sequence dn it is convenient to deal with maps F̃ such that (F̃ n )∗ = (F̃ ∗ )n . (9) Maps F̃ satisfying condition (9) are called Algebraically Stable maps (AS for short), (see [18]). 10 In order to get AS maps we will use the following useful result showed by Fornaess and Sibony in [19] (see also Theorem 1.14) of [18]: The map F̃ is AS if and only if for every exceptional curve C and all n ≥ 0 , F̃ n (C) ∈ / I(F̃ ). (10) It is known (see Theorem 0.1 of [18]) that one can always arrange for a birational map Pk−1 to be AS considering an extension of f. If it is the case and we call X (x) = xk + i=0 ci xi the characteristic polynomial of A := (F̃ ∗ ), then since X (A) = 0 and di is the (1, 1) term of Ai we get that dk = −(c0 + c1 d1 + c2 d2 + · · · + ck−1 dk−1 ), i. e., the sequence dn satisfies a homogeneous linear recurrence with constant coefficients. The dynamical degree is then the largest real root of X (x). The following result is useful in our work. It is a direct consequence of Theorem 0.2 of [18]. Given a birational map F of P C2 , let F̃ be its regularized map so that the induced map F̃ ∗ : Pic(X) → Pic(X) satisfies (F̃ n )∗ = (F̃ ∗ )n . Then Theorem 6. (See [18]) Let F : P C2 → P C2 be a birational map, F̃ be its regularized map and let dn = deg(F n ). Then up to bimeromorphic conjugacy, exactly one of the following holds: • The sequence dn grows quadratically, F̃ is an automorphism and f preserves an elliptic fibration. • The sequence dn grows linearly and f preserves a rational fibration. In this case F̃ cannot be conjugated to an automorphism. • The sequence dn is bounded, F̃ is an automorphism and f preserves two generically transverse rational fibrations. • The sequence dn 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. 3.3 Lists of orbits. We derive our results in the non-degenerate case by using Theorem 7 below, established and proved in [3]. 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. 11 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 (1), 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 C2 as follows: J[x0 : x1 : x2 ] = [x1 x2 : x0 x2 : x0 x1 ]. We find that the involution J has an indeterminacy locus I = {0 , 1 , 2 } and a set of T exceptional curves E = {Σ0 , Σ1 , Σ2 }, where Σi = {xi = 0} for i = 0, 1, 2, and i = Σj Σk with {i, j, k} = {0, 1, 2} and i 6= j 6= k , i 6= k. Let I(G−1 ) := {a0 , a1 , a2 }, the elements of this set are determined by ai := G(Σi − I(J)) = L i for i = 0, 1, 2; see [3]. To follow the orbits of the points of I(G−1 ) we need to understand the following defini- tions and construction of lists of orbits in order to apply the result of Theorem 7. We assemble the orbit of a point p ∈ P C2 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 C2 such that its nth − 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), ..., Gn (p)} and we call it a singular orbit. If for some p ∈ P C2 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), G2 (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 Gn (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 7 we need to organize our SE orbits into lists in the following way. Two orbits O1 = {a1 , ..., j1 } and O2 = {a2 , ..., j2 } are in the same list if either j1 = 2 or j2 = 1, that is, if the ending index of one orbit is the same as the beginning index of the other. We have the following possibilities: • Case 1: One SE-orbit, Oi = {ai , ..., τ (i) }. Then we have the list L = {Oi = {ai , ..., τ (i) }}. If τ (i) = i we say that L is a closed list. Otherwise it is an open list. • Case 2: Two SE-orbits, Oi = {ai , ..., τ (i) }} and Oj = {ai , ..., τ (j) }}. In this case we can have either two closed lists, L1 = {Oi = {ai , ..., i }} and L2 = {Oj = {aj , ..., j }} with i 6= j or one open and one closed list L1 = {Oi = {ai , ..., i }} and L2 = {Oj = {aj , ..., k }} with i 6= j , j 6= k , k 6= i 12 or a single list L = {Oi = {ai , ..., j }, Oj = {aj , ..., τ (j) }} with i 6= j which is closed if τ (j) = i and an open list otherwise. Notice that we cannot have two open lists because there are at most three SE-orbits. • Case 3: Three SE orbits: In this case we can have either three closed lists L1 = {O0 = {a0 , ..., 0 }} and L2 = {O1 = {a1 , ..., 1 }} and L3 = {O2 = {a2 , ..., 2 }}, or two closed lists L1 = {Oi = {ai , ..., j }, Oj = {aj , ..., i }} and L2 = {Ok = {ak , ..., k }} with i 6= k 6= j and i 6= j or one closed list L = {O0 = {a0 , ..., 1 }, O1 = {a1 , ..., 2 }, O2 = {a2 , ..., 0 }}. We now define two polynomials TL and SL which we will use to state theorem 7. Let ni denote the sum of the number of elements of an orbit Oi and let NL = nu + ... + nu+µ denote the sum of the numbers of elements of each list |L| . If L is closed then TL = xNL − 1 and if L is open then TL = xNL . Now we define SL for different lists as follows:    1 if |L| = {n1 } ,     n n x 1 +x 2 +2 if L is closed and |L| = {n1 , n2 } ,      n n x 1 +x 2 +1 if L is open and |L| = {n1 , n2 } , SL (x) = P  N −ni 3    x L + xn i + 3 if L is closed and |L| = {n1 , n2 , n3 } ,     i=1   P 3 P ni   xNL −ni + x +1 if L is open and |L| = {n1 , n2 , n3 } .  i=1 i6=2 Theorem 7. ([4]) If G = L ◦ J, then the dynamical degree δ(G) is the largest real zero of the polynomial Y X Y X (x) = (x − 2) TL (x) + (x − 1) SL (x) TL0 (x). L∈Lc ∪Lo L∈Lc ∪Lo L0 6=L Here L runs over all the orbit lists. 13 4 Proof of the results F is a non-degenerate when the sets E(F ), E(F −1 ) have the three elements, each two of them intersecting on distinct points of I(F ), I(F −1 ) presented in section 2.1. In this section we consider this to do the following study. We consider the involution J[x0 : x1 : x2 ] introduced in section 2.3, and two invertible linear maps M1 and M2 of P C2 such that M1 sends each Σi to Si and M2 (Ai ) = i for i = 0, 1, 2. Then the mapping M2 ◦ F ◦ M1 is quadratic and sends each Σi to i . Therefore M2 ◦ F ◦ M1 must be of the form [λ0 x1 x2 : λ1 x0 x2 : λ2 x0 x1 ] with λi 6= 0 for each i = 0, 1, 2, that is M2 ◦ F ◦ M1 = D ◦ J where D=diag(λ0 , λ1 , λ2 ). Calling L = M1−1 ◦ M2−1 ◦ D we get that F ◦ M1 = M2−1 ◦ D ◦ J = M1 ◦ L ◦ J, that is the mappings F and G := L ◦ J are linearly conjugated. Calling ai := G(Σi − I(J)) = Li for i = 0, 1, 2 we are going to identify each ai ∈ I(G−1 ) with Ai ∈ I(F −1 ). From now on we are going to assume that f (x, y) is birational (see conditions (a) in Lemma 5), that F [x0 : x1 : x2 ] has degree two and that it is not not degenerated (i. e., (αγ)12 6= 0 6= (βγ)12 ). The exceptional set of F and F −1 can be seen in the following figure: The mapping F is bijective from P C2 \{S0 , S1 , S2 } to P C2 \{T0 , T1 , T2 }. The only points in Ti which have preimage by F are Aj , Ak with i ∈ / {j, k} , j 6= k which have as preimages Sj and Sk respectively. To prove the results we use the following strategy. First we perform the necessary blow- up’s in order to have an extension of F, that is F̃ and it is AS. Then we construct the lists of the orbits of points Ai and we apply Theorem 7. We now give the proofs of Theorems 1 to 4. They are as follows. 1. Proof of Theorem 1 Proof. The conditions on the parameters imply that F (A0 ) = A0 with A0 ∈ / I(F ) and F (A1 ) = A0 . Since A0 , A1 ∈ S0 ∈ E(F ), thus we find that their orbits are O0 = {A0 } and O1 = {A1 } which are singular but not elementary. Now it remains to analyze the behavior of iterates of A2 . We claim that @p ∈ N : F p (A2 ) = O1 and @p ∈ N : F p (A2 ) = O2 . It is so because if F p (A2 ) = O1 then, since O1 ∈ S0 = T2 it would imply O1 = A0 or O1 = A1 , that is α1 = 0 or α2 = 0. Similarly, F p (A2 ) = O2 implies γ1 = 0 or γ2 = 0. Therefore the only possibility is that any iterate of A2 reaches O0 . Thus we assume the following cases: (a) Assume that F p (A2 ) 6= O0 for all p ∈ N. Then the map F is itself AS. Hence, δ(f ) = 2. 14 (b) Now assume that F p (A2 ) = O0 for some p ∈ N. Thus we have a SE orbit of A2 which is: O2 = {A2 , F (A2 ), ..., F p (A2 ) = O0 }. In this case we have only one list Lo which is open. That is: Lo = {O2 = {A2 , F (A2 ), ..., F p (A2 ) = O0 }}. To find the characteristic polynomial we use Theorem 7. We find that NLo = p, TLo = xp and SLo = 1. Then the δ(F ) is the largest root of the characteristic polynomial Yp (x) := xp+2 − 2xp+1 + x − 1. Observe that for all the values of p ∈ N the above polynomial has always the largest root λ > 1. This is because Yp (1) = −1 < 0 and Yp (2) = 1 > 0, therefore there always exists a root λ > 1 such that Yp (λ) = 0. Hence dn has exponential growth rate. 2. Proof of Theorem 2 Assume that γ1 = 0. From Lemma 5 we know that α1 , β1 and γ2 are non zero. We distinguish two cases, depending on α2 . • Consider the case when α2 = 0. Observe that S0  A0 = O2 and S1  A1 = O1 . Hence we blow up the points A0 , A1 to get the exceptional fibres E0 , E1 . Let X be the new space and let F̃ : X → X be the extended map on X. In order to know F̃ we see [u : v]E0 ∈ S0 (resp. [u : v]E1 ∈ S0 ) as limt→0 [tu : 1 : tv] (resp. limt→0 [tu : tv : 1]), we evaluate F [tu : 1 : tv] (resp. F [tu : tv : 1]) and take limits again. We get: F̃ [0 : x1 : x2 ] = [x2 : x1 + βx2 ]E0 , F̃ [u : v]E0 = [0 : α1 v : u] ∈ T2 = S0 and F̃ [x0 : x1 : − γγ20 x0 ] = [x0 : α0 x0 + α1 x1 ]E1 F̃ [u : v]E1 = [γ2 u : γ2 (α0 u + α1 v) : β2 u] ∈ T1 Then the map F̃ sends the curve S0 → E0 → S0 and S1 → E1 → T1 . We observe that no new point of indeterminacy is created therefore I(F̃ ) = {O0 } and E(F̃ ) = {S2 } . Assume that there exists p ∈ N such that F̃ p (A2 ) = O0 . Then we blow up A2 , F̃ (A2 ), F˜2 (A2 ), . . . , F˜p (A2 ) = O0 getting the exceptional fibres which we call E2 , E3 , . . . , Ep+2 . Set F˜1 : X1 → X1 the extended map. Performing the blow up at O0 , since T0 is sent to O0 via F −1 , we have that F̃1−1 : T0 → Ep+2 . 15 Then S2 → E2 → E3 → · · · → Ep+1 → Ep+2 → T0 . Hence F̃1 : X1 → X1 is an AS map and also an automorphism. Now we have two closed lists as follows Lc1 = {O0 = {A0 = O2 }, O2 = {A2 , F̃ (A2 ) , . . . , F̃ p (A2 ) = O0 }}, Lc2 = {O1 = {A1 = O1 }}. Then by using Theorem 7 we find that the characteristic polynomial associated to F is X = (xp+1 +1)(x−1)2 (x+1). If p is even then xp+1 +1 has the factor x+1 and X = (x − 1)2 (x + 1)2 (xp − xp−1 + · · · − x + 1). Hence the sequence of degrees is dn = c0 + c1 n + c2 (−1)n + c3 n (−1)n + c4 λn1 + c5 λn2 + ... + cp+3 λnp , where ci are constants and λ1 , λ2 , ..., λp are the roots of polynomial xp − xp−1 + · · · − x + 1. By looking at dn we see that f does not grow quadratically or exponentially. As our map F̃1 is an automorphism then by using the results from Diller and Favre in [18] we see that also cannot have linear growth. Therefore we must have c1 = c3 = 0. Hence the sequence of degrees must be periodic. This implies that d2p+2+n = dn i.e. the sequence of degrees is periodic with period 2p + 2. If p is odd then dn is also periodic of period 2p + 2. If F̃ p (A2 ) 6= O0 for all p ∈ N, then we have two lists which are open and closed as follows: Lo = {O0 = {A0 = O2 }} , Lc = {O1 = {A1 = O1 }}. Then δ(F ) is determined by the polynomial (x − 1)2 (x + 1), and δ(f ) = 1. The 5 sequence of degrees is dn = 4 + 12 n − 41 (−1)n . • Now consider that α2 6= 0. The parameters α1 , β1 , γ2 are all non zero. Observe that S0  A0 = O2 . The orbit of A0 is SE. By blowing up A0 we get the exceptional fibre E0 and the new space X. The induced map F̃ : X → X sends the curve S0 → E0 → S0 . Observe that now I(F̃ ) = {O0 , O1 } and E(F̃ ) = {S1 , S2 } . We see that A1 6= O1 and the exceptional curve S1  A1 ∈ S0 . We observe that the collision of orbits discussed in preliminaries is happening here. The orbit of A1 under F̃ is as follows: S1  A1 → [γ2 : β2 ]E0 → [0 : α1 (γ0 + β2 ) : β1 ] ∈ S0 → · · · After some iterates we can write the expression of F̃ 2k (A1 ) for all k > 0 ∈ N as F̃ 2k (A1 ) = [0 : α1 (γ0 + β2 )(1 + α1 + α12 + · · · + α1k−1 ) : β1 ] ∈ S0 . Observe that for some value of k ∈ N it is possible that F̃ 2k (A1 ) = O1 . This happens when the following condition k is satisfied for some k. α12 (γ0 + β2 )(1 + α1 + α12 + · · · + α1k−1 ) + α2 β1 = 0. (11) 16 For such k ∈ N the orbit of A1 is SE. By blowing up the points of this orbit we get the new space X1 and the induced map F̃1 . Then under the action of F̃1 we have S1 → G0 → G1 → G2 → · · · → G2k−1 → G2k → T1 . Then I(F̃1 ) = {O0 } and E(F̃1 ) = {S2 }. Now if the orbit of A1 is SE and if F̃1p (A2 ) = O0 that is the orbit of A2 is also SE for some p ∈ N then we have three SE orbits. If condition k is not satisfied then with the extended map F̃ we have I(F̃ ) = {O0 , O1 }. Therefore we have two options: F̃ p (A2 ) = O0 or F̃ p (A2 ) = O1 . We claim that for all p ∈ N, F̃ p (A2 ) 6= O1 . Assume that F̃ p (A2 ) = O1 and assume that F j (A2 ) ∈ / S0 for j = 1, 2, . . . , p − 1. F̃ p (A2 ) = F p (A2 ) = O1 . Since / S0 if F p (A2 ) = O1 then p would be greater than zero and O1 ∈ S0 and A2 ∈ since S0 = T2 , it would imply that O1 = A1 or O1 = A2 , which is not the case (recall that the only points in T2 which have a preimage are A1 and A2 ). Contrarily, if it exists some l ∈ N, l < p such that F j (A2 ) ∈ / S0 for j = 1, 2, . . . , l− 1 but F l (A2 ) ∈ S0 \ {O1 } then F l (A2 ) must be equal to A1 or A2 that is, F l (A2 ) = A1 or F l (A2 ) = A2 . The second case is not possible as A2 is a fixed point. In the first case F̃ p (A2 ) = F̃ p−l (F l (A2 )) = F̃ p−l (A1 ) = O1 which implies that p = l + 2r and F̃ 2r (A1 ) = O1 . Hence the orbit of A1 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 O2 to be SE is to have that for some p, F̃ p (A2 ) = O0 . After the blow up process we get S2 → E1 → E2 → · · · → Ep → Ep+1 → T0 . The extended map F̃2 is an automorphism when we have three SE orbits. The above discussion gives us three different cases. – One SE orbit: This happens when A0 = O2 with the conditions that F̃ 2k (A1 ) 6= O1 and F̃ p (A2 ) 6= O0 for all k, p ∈ N. Therefore we have on- ly one list Lo which is open that is Lo = {O0 = {A0 = O2 }}. By using √ 5+1 theorem 7 we find that δ(F ) = δ ∗ = 2 . which is given by the greatest root of the polynomial X(x) = x2 − x − 1. Therefore it has exponential growth. – Two SE orbits (a): It is the case when A0 = O2 , F̃ p (A2 ) = O0 and F̃ 2k (A1 ) 6= O1 for all k ∈ N. By organizing the orbits into lists we have one closed list Lc = {O0 = {A0 = O2 }, O2 = {A2 , F̃ (A2 ) , ..., F̃ p (A2 ) = O0 }}. 17 By utilizing theorem 7 we find that the characteristic polynomial associated to F is Xp = xp+1 (x2 − x − 1) + x2 . For p = 0 and p = 1 the sequence of degrees satisfies dn+3 = dn and dn+4 = dn+3 respectively which corresponds towards boundedness of f . For p = 2 we get the polynomial X2 = x2 (x + 1)(x − 1)2 . Looking at the first degrees we get that the sequence of degrees is dn = −1 + 2 n. For p > 2, we observe that Xp (1) = 0, X 0 p (1) = 2−p < 0 and limx→+∞ Xp (x) = +∞. Hence Xp always has a root λ > 1 and the result follows. 2k p – Two SE orbits (b): When we have A0 = O2 , F˜1 (A1 ) = O1 and F˜1 (A2 ) 6= O0 for all p ∈ N then there is one open and one closed list and Xk = x2k+1 (x2 − x − 1) + 1. We observe that for all the values of k ∈ N , k ≥ 1 the polynomial Xk has always a root λ > 1. Therefore f has exponential growth. – Three SE orbits: In this case we have A0 = O2 , F̃ 2k (A1 ) = O1 , F̃ p (A2 ) = O0 , for a certain p, k ∈ N. We have two closed lists as follows: Lc = {O0 = {A0 = O2 }, O2 = {A2 , F̃ (A2 ) , ..., F̃ p (A2 ) = O0 }}, Lc = {O1 = {A1 , F̃ (A1 )E0 , ..., F̃ 2k (A1 )S0 = O1 }}. From theorem 7 we can write X(k,p) = xp+1 (x2k+3 − x2k+2 − x2k+1 + 1) + x2k+3 − x2 − x + 1. The map F̃2 is an automorphism for all the values (k, p). According to Diller and Favre in [18] the growth of degrees 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. we consider it’s Taylor expansion near x = 1 : X(k,p) (x) = 2(2 − kp + 2k)(x − 1)2 + O(|x − 1|3 ). 2(1+k) Thus X(k,p) vanishes at x = 1 x = 1 and has a maximum on it p > k . Since limx→+∞X(k,p) (x) = +∞, always exists a root greater than one. If 2(1+k) p ≤ k , k ≥ 1 then the pairs (k, p) are in the set: A(k,p) = {((k ≥ 1), 0), ((k ≥ 1), 1), ((k ≥ 1), 2), (1, 3), (2, 3), (1, 4)}. For (k, p) = (k, 0), when k is even the sequence of degrees is dn = c0 + c1 n + c2 (−1)n + c3 (−1)n n + c4 λn1 + c5 λn2 + ... + c2k+3 λn2k , where ci are constants and λ’s are the roots of polynomial x2k+2 = 1 different from ±1. If k is odd then dn = l0 + l1 n + l2 (−1)n + l3 µn1 + l4 µn2 + · · · + l2k+3 µn2k+1 , 18 where li are constants and µ’s are the roots of polynomial (xk+1 − 1) (xk + xk−1 + · · · + x + 1). Since F̃2 is an automorphism for all (k, p), using [18] we have c1 = 0 = c3 and also l1 = 0. This implies that d2k+2+n = dn , i. e., the sequence of degrees is periodic with period 2k + 2. The argument for the proof of other values of (k, p) ∈ A(k,p) follows accordingly. 3. Proof of Theorem 3 From hypothesis and from Lemma 5 we know that α2 γ1 6= 0 and β2 γ1 6= 0 therefore α2 , β2 and γ1 cannot be zero. There exist two different cases to study depending on α1 . • Consider that α1 = 0. Then A0 = O1 and A1 = O2 . We get new space X by blowing up A0 and A1 . E0 , E1 are the exceptional fibres on these points respectively. The extended map F̃ : X → X sends S0 → E0 → T1 and S1 → E1 → T2 . Therefore the orbits of A0 and A1 are SE. No new indeterminacy points have appeared therefore I(F̃ ) = {O0 } and E(F̃ ) = {S2 } . If F̃ p (A2 ) = O0 for some p ∈ N then the orbit of A2 is SE. Let F̃2 : X1 → X1 be the extended map on new space X1 we get after blowing up the points of orbit of A2 . Then F̃2 sends S2 → E2 → E3 → · · · → Ep+2 → T0 . Then F̃2 is an AS map and is an automorphism. We see that we have one closed list and by utilizing Theorem 7 the characteristic polynomial associated to F is Yp = xp+1 (x3 − x − 1) + (x3 + x2 − 1). For p = 0, the sequence of degrees dn = c1 + c2 (−1)n + c3 (λ1 )n + c4 (λ2 )n , where λ1 , λ2 are the two roots of x2 + x + 1 = 0. Hence dn satisfies dn+6 = dn , i.e., it is periodic of period 6. For p ≤ 5 the argument for the proof is similar with periods 5, 8, 12, 18 and 30 accordingly. When p = 6, the sequence of degrees dn = c1 + c2 n + c3 n2 + c4 (−1)n + c5 (λ1 )n + c6 (λ2 )n + c7 (λ3 )n + c8 (λ4 )n + c9 (λ5 )n + c10 (λ6 )n . As F̃2 is an automorphism, from Theorem 6, the sequence of degrees does not grow linearly. Then either, c3 6= 0 and dn grows quadratically or c2 = 0 = c3 and dn is periodic of period 30. For p > 6 there always exists a root λ > 1 of Yp . Hence the sequence of degrees grows exponentially for such values of p. Now suppose that no p exists such that F̃ p (A2 ) = O0 . In this case δ(F ) is given by the greatest real root of the polynomial Y(x) = x3 − x − 1. • Now consider that α1 6= 0. Observe that in general S0  A0 6= Oi for any i ∈ {0, 1, 2} and F (A0 ) = A0 . Thus O0 = {A0 } is not a SE orbit. Now S1  A1 = O2 . We blow up the point A1 = O2 . Therefore the orbit of A1 is SE. Let 19 X be the new space after blowing up A1 and let E1 be the exceptional fibre at this point. The induced map F̃ : X → X sends the curve S1 → E1 → T2 = S0 . Then I(F̃ ) = {O0 , O1 } and E(F̃ ) = {S0 , S2 } . The curve S2  A2 and O1 ∈ S0 = T2 . Note that F p (A2 ) 6= O1 . As the only points on T2 which have preimages are A0 and A1 . Then if the orbit of A2 reaches O1 at some iterate of F then O1 should be equal to either A0 or A1 . As α1 6= 0 hence A0 6= O1 6= A1 . This implies that F p (A2 ) 6= O1 for all p but it is possible that F̃ p (A2 ) = O1 . Then in general there are two possibilities: F̃ p (A2 ) = O0 for some p ∈ N or F̃ q (A2 ) = O1 for some q ∈ N. In both cases the orbit of A2 is SE. Now if there exists some p ∈ N such that F̃ p (A2 ) = O0 , then to get X1 we blow-up all the points of the orbit of A2 . The extended map F̃1 : X1 → X1 sends S2 → E2 → E3 → · · · → Ep+2 → T0 . This shows that F̃1 is an AS map. Now we have one open list and the characteristic polynomial associated to F is Yp = xp+1 (x2 − x − 1) + x2 − 1. Observe that for all the values of p ∈ N the polynomial Yp always has the largest root λ > 1. Hence dn grows exponentially √ 1+ 5 and δ(F ) approaches to the value δ ∗ = 2 as p → ∞. Also if there exists some q ∈ N such that F̃ q (A 2) = O1 , then F̃1 is an AS map. We have one closed list and the characteristic polynomial associated to F is Yq = xq+1 (x2 − x − 1) + x2 . Note that there are no mappings for q ∈ {0 , 1}. As A2 [1] 6= O1 [1] therefore q = 0 is not possible. For q = 1 we have two possibilities. First when A2 ∈ / S1 , then the condition F̃ (A2 ) = F (A2 ) = O1 . But the orbit of A2 can never reach O1 because O1 ∈ T2 and O1 6= A0 . Now if A2 ∈ S1 then we have the condition F̃ (A2 ) = O1 . In this case F̃ (A2 ) ∈ E1 but it is clear that O1 ∈ / E1 therefore q = 1 is not possible. For q = 2 we get the polynomial x2 (x + 1)(x − 1)2 and the sequence of degrees is dn = c2 (−1)n + c3 + c4 n. Looking at the first degrees we get dn = −1 + 2n. For q > 2, we observe that Yq always has a root λ > 1 and the result follows. 4. Proof of Theorem 4 Considering the hypothesis we know that γ1 6= 0 6= γ2 therefore from lemma 5 the parameters {α0 , β0 , β1 , β2 , γ0 } and one of {α1 , α2 } at the same moment can be zero. Hence two different cases, α1 = 0 and α2 = 0 are considered as follows: • Consider that α1 = 0 and α2 6= 0. Observe that S0  A0 = O1 . Let X be the new space we get after blowing up the point A0 and let E0 be the exceptional fibre at this point. The induced map F̃ : X → X sends the curves S0 → E0 → T1 . Hence I(F̃ ) = {O0 , O2 } 20 and E(F̃ ) = {S1 , S2 } . Note that A1 is not indeterminate for F and A1 ∈ S0 therefore orbit of A1 collides with A0 . But we observe that F̃ q (A1 ) 6= O2 for all q as O2 ∈ S0 = T2 . However it is possible that F̃ p (A1 ) = O0 for some p ∈ N. If there exists such p then we blow up the points of the orbit of A1 to get the exceptional fibres Ei ’s. Let X1 be the new space. Then the extended map F̃1 sends S1 → E1 → E2 → · · · → Ep+1 → T0 . Now I(F̃1 ) = {O2 } and E(F̃1 ) = {S2 } . The exceptional curve S2  A2 . But the orbit of A2 can never reach O2 as O2 ∈ T2 . Thus the orbit of A2 is not SE. Hence the map F̃1 is AS. In this case we have one closed list by using Theorem 7 the characteristic poly- nomial associated to F is Zp = xp+1 (x2 − x − 1) + x2 . The proof for all values of p is similar to the last part of theorem 3. Now assume that F̃ p (A1 ) 6= O0 for all p ∈ N such that the orbit of A1 is not SE then for some q it is possible that F̃ q (A2 ) = O0 . In this case after the blow up of the orbit of A2 , in the extended space X1 the induced map F̃1 acts as S2 → E1 → E2 → · · · → Eq+1 → T0 . Thus the orbit of A2 is SE and the map F̃1 is AS. We have one open list and the characteristic polynomial associated to F is Zq = xq+1 (x2 −x−1)+x2 −1. We observe that for all the values of q ∈ N the polynomial Zq always the largest root λ > 1 and dn grows exponentially. We now consider the case when for some p, q ∈ N we have F̃ p (A1 ) = O0 and F̃ q (A2 ) = O0 . We claim that p must be different from q. Assume that F̃ k (A1 ) 6= A2 and F̃ j (A2 ) 6= A1 for any 0 < k < p and 0 < j < q. Because otherwise these points can have multiple preimages. Then p = q gives the condition that F̃ p (A1 ) = F̃ p (A2 ) = O0 implies that A1 = A2 , as F is bijective except for some particular points. But this gives a contradiction as all Ai ’s must be dif- ferent in this case. Now if there exists some k or j such that F̃ k (A1 ) = A2 or F̃ j (A2 ) = A1 then there is collision of orbits. This gives that either, k > 0 or j > 0 which shows that p 6= q. Now consider that q > p. Then the orbit of A2 must collides with the orbit of A1 . Because if not then this claims that F̃ k (A2 ) 6= A1 for all 0 < k < q. As q > p then for some j > 0 we can write q = p + j. This gives F̃ q (A2 ) = F̃ j+p (A2 ) = O0 = F̃ p (A1 ). As O0 has unique preimage and there is no collision this implies that no orbit enters any T1 or T2 . Therefore the points F̃ j+p (A2 ) and F̃ p (A1 ) have unique preimages. Then for F̃ j+p (A2 ) = O0 = F̃ p (A1 ) we can find the preimages by iterating p times with F̃ −1 . This gives us F̃ −p (F̃ j+p (A2 )) = F̃ −p (F̃ p (A1 )) 21 which implies that F̃ j (A2 ) = A1 for some 0 < j < q, which gives contradiction to our claim. This implies that in the case when q > p or q < p we always have collision of orbits of A2 with A1 or A1 with A2 . Now for q > p we must have F̃ k (A2 ) = A1 for some 0 < k < q. Then we see that: S2  A2 → F̃ (A2 ) → · · · → F̃ q−p (A2 ) = A1 → F̃ (A1 ) → · · · → F̃ p (A1 ) = O0 . This implies that the orbit of A2 is no more SE. Hence two SE orbits are the orbits of A0 and A1 . This shows that the characteristic polynomial is Zp in this case. Similarly, in the second case the characteristic polynomial is Zq . Finally, if F̃ p (A1 ) 6= O0 and F̃ p (A2 ) 6= O0 for any p ∈ N then we have one SE orbit and the characteristic polynomial is given by Z(x) = x2 − x − 1. Then the dynamical degree δ(f ) = δ ∗ . • Now consider α2 = 0, from Lemma 5 we have α1 , γ1 , γ2 non zero. Observe that A0 is not an indeterminate for F and is a fixed point of F. Hence the orbit of A0 is not SE. Now S1  A1 = O1 . After blowing up the point A1 to get the exceptional fibre E1 , the induced map F̃ : X → X for new space X sends the curve S1 → E1 → T1 . Now I(F̃ ) = {O0 , O2 } and E(F̃ ) = {S0 , S2 } . The exceptional curve S2  A2 . Note that F̃ p (A2 ) 6= O2 for all p ∈ N as O2 ∈ T2 . Then for some p ∈ N it is possible that F̃ p (A2 ) = O0 . In this case the orbit of A2 is SE. Let X1 be the expanded space we get after blowing up the orbit of A2 the observe that now F̃1 : X1 → X1 is an AS map but is not an automorphism as S0 still collapses. In this case we have one closed list and one open and the characteristic polynomial associated to F is Zp = (xp+1 + 1)(x − 1)2 . Now if no such p exists so that F̃ p (A2 ) = O0 then we have one open list δ(F ) is given by the largest root of the polynomial Z(x) = (x − 1)2 , which is one. 5 Zero entropy cases From the results of Theorems 1, 2, 3 and 4 we can present the maps with zero algebraic entropy. Looking at Theorem 1 we see that the maps which satisfy their hypothesis have not zero entropy. From Theorem 2 we find very interesting maps with zero entropy. This case is studied in detail in the paper [9], giving all the prescribed invariant fibrations and also recognizing which of such a maps are periodic and/or integrable. 22 Among the mappings satisfying the hypothesis of Theorem 3 we have the ones with α1 = 0. After an affine change of coordinates this maps can be written as   β0 + y f (x, y) = y, . γ0 + x These are the maps which we deal when we want to study a linear fractional recurrence of order two, and they are analized in paper [4]. When α1 6= 0 the only case with dynamical degree equals one is when F̃ 2 (A2 ) = O1 , where F̃ is the mapping induced by F after blowing up the point A1 . To find the maps with this condition in principle we have two possibilities, / S1 then the condition F̃ 2 (A2 ) = F 2 (A2 ) = O1 with or without collision of orbits. When A2 ∈ never is satisfied. It is because O1 ∈ S0 = T2 and the only points on T2 which have preimage by F are A0 and A1 and we see that A0 6= O1 6= A1 . If A2 ∈ S1 , that is if β0 = α0 , then F̃ 2 (A2 ) = F̃ (F̃ ([1 : 0 : −α1 ])) = F̃ ([1 : α0 − α1 ]E1 ) = [0 : α0 − α1 : 1]. α21 −1 Hence, condition F̃ 2 (A2 ) = O1 = [0 : 1 : −α1 ] is satisfied for α0 = α1 = β0 and we get the uniparametric family of mappings   ω+y α2 − 1 f (x, y) = ω + α1 x + y, , ω= 1 , α1 6= 0. x α1 Since the corresponding sequence of degrees grows linearly, f preserves a unique rational fibration. In fact, (1 + α1 x)(α1 + α1 x + y) V (x, y) = x satisfies V (f (x, y)) = α1 V (x, y). When α1 = 1 for some n ∈ N then defining W (x, y) = n V (x, y)n we see that W (f (x, y)) = W (x, y), that is f is integrable. We observe that we also know that these maps never are periodic maps as the degrees grow linearly. In a similar way, we find   β2 y f (x, y) = α2 y, −α2 β2 + x + y which satisfies the hypothesis of Theorem 4 with α1 = 0 and has the unique invariant fibration (β2 − y)(α2 β2 − x) V (x, y) = y with the property V (f (x, y)) = −α2 V (x, y). As before when (−α2 )n = 1, then W (x, y) = V (x, y)n is a first integral of f (x, y). Finally, if f (x, y) satisfies the hypothesis of Theorem 4 with α1 6= 0 and has zero entropy, after an affine change of coordinates can be written as   β0 + y f (x, y) = α0 + α1 x, . x+y 23 Clearly V (x, y) = x is an invariant fibration satisfying V (f (x, y)) = α0 + α1 V (x, y). When α1n = 1 6= α1 then calling h(x) := α0 + α1 x we have that W (x, y) = x h(x) h2 (x) . . . hn−1 (x) is a first integral of f (x, y). Also when α1 = 1 and α0 = 0 the maps are integrable. Acknowledgements The first author is supported by Ministry of Economy, Industry and Competitiveness of the Spanish Government through grant MINECO/FEDER MTM2016-7 7278-P and also by the grant 2014-SGR-568 from AGAUR, Generalitat de Catalunya. References [1] Angles, JC, Maillard, JM and Viallet, C. On the complexity of some birational trans- formations, J. Phys. A: Math. Gen. 39 (2006) 36413654. 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