Studies on spaces of initial conditions for nonautonomous mappings of the plane
aa r X i v : . [ m a t h - ph ] F e b STUDIES ON SPACES OF INITIAL CONDITIONS FORNONAUTONOMOUS MAPPINGS OF THE PLANE
TAKAFUMI MASE
Abstract.
We study nonautonomous mappings of the plane by means of spaces ofinitial conditions. First we introduce the notion of a space of initial conditions fornonautonomous systems and we study the basic properties of general equations thathave spaces of initial conditions. Then, we consider the minimization of spaces of initialconditions for nonautonomous systems and we show that if a nonautonomous mappingof the plane with a space of initial conditions, and unbounded degree growth, has zeroalgebraic entropy, then it must be one of the discrete Painlev´e equations in the Sakaiclassification. Introduction
Mappings of the plane are among the main objects of interest in the field of discreteintegrable systems. Such a mapping ϕ n : ( x n , y n ) ( x n +1 , y n +1 )can be thought of as defining the equation( x n +1 , y n +1 ) = ϕ n ( x n , y n ) , where x n +1 and y n +1 are functions of x n and y n (and n ). Hereafter we shall thereforeoften refer to such a mapping as an equation itself. A three point mapping, in which x n +1 is determined by x n and x n − , can be transformed to the above form by introducing y n = x n +1 .In this paper, we deal with mappings of the plane that can be rationally solved in theopposite direction. Such an equation defines a (family of) birational automorphism(s) on P (or on P × P ).How to detect the integrability of discrete equations has been a major problem in thefield of integrable systems for more than a quarter century.Singularity confinement was first proposed by Grammaticos, Ramani and Papageorgiou[15] as a discrete analogue of the Painlev´e property in continuous systems. Where thePainlev´e property requires all movable singularities to be at most poles, singularity con-finement requires every singularity (i.e. disappearance of information on the initial values)to be confined after a finite number of iterates. An equation is said to “enter a singularity”when loosing information on the initial values, and is said to “exit from a singularity”when recovering the lost information. Singularity confinement is so powerful that manydiscrete Painlev´e equations have been discovered by deautonomising QRT mappings solelywith the help of singularity confinement [14].However, Hietarinta and Viallet presented [20] an equation that passes the singularityconfinement test but which exhibits chaotic behavior. Their counterexample is(1.1) x n +1 + x n − = x n + ax n , hich is now called the Hietarinta-Viallet equation. In order to test for integrability moreprecisely, Bellon and Viallet defined the algebraic entropy [4] and showed that the entropyof the above equation is log((3 + √ / > Definition 1.1 (algebraic entropy [4], dynamical degree) . The limitslim n →∞ n log (deg ϕ n ) and lim n →∞ (deg ϕ n ) /n , if they exist, are called the algebraic entropy and the dynamical degree of the equation,respectively. We denote by ϕ n the n -th iterates and by deg ϕ n the degree of ϕ n as arational function of the initial values in the iteration.It is obvious that the entropy coincides with the logarithm of the dynamical degree.Even in the autonomous case (“autonomous” meaning that ϕ does not depend on n ),it is difficult to calculate the exact value of the entropy for a concrete equation. However,the integrability test based on zero algebraic entropy is empirically accurate. Hereafter,we shall call an equation with zero algebraic entropy integrable. Remark 1.2.
It is known that in the autonomous case, the entropy exists in nonneg-ative real number and that is invariant under coordinate changes [4]. However, as inExample 2.2, this does not hold in the nonautonomous case.
Remark 1.3.
There are several definitions for the degree of a mapping of the plane.The degree as a birational automorphism on P (Definition A.6) is the most standardone. We will mainly use this degree in this paper.If ϕ is written as ϕ ( x, y ) = (cid:18) ϕ ( x, y ) ϕ ( x, y ) , ϕ ( x, y ) ϕ ( x, y ) (cid:19) , where ϕ i and ϕ i have no common factors for i = 1 ,
2, then the degree of ϕ as a birationalautomorphism on P × P is defined bydeg ϕ = max(deg ϕ , deg ϕ , deg ϕ , deg ϕ ) . This degree is particularly convenient when we consider three point mappings.It is known that, while these two degrees are different, their growth as a function of n is the same. Example 1.4.
Consider the equation ϕ ( x, y ) = (cid:18) y , x (cid:19) . It immediately follows from the above expression that the degree of ϕ as a birationalautomorphism on P × P is 1.On the other hand, ϕ can be written in homogeneous coordinates on P as ϕ ( z : z : z ) = ( z z : z z : z z ) . Therefore, the degree of ϕ as a birational automorphism on P is 2.Since ϕ = id, the degree growth of ϕ n is bounded in both cases.In the nonautonomous case, the degree of the n -th iterate of a mapping ϕ isdeg ϕ n = deg( ϕ ℓ + n − ◦ · · · ◦ ϕ ℓ ) , which in general depends on the starting index ℓ . However, we rarely think of deg ϕ n (orthe entropy) as a function of ℓ . It is usual to fix the starting index (for example ℓ = 0, as n Example 2.2) or only consider these cases where deg ϕ n does not depend on ℓ , for all n .If so, then the algebraic entropy always exists for the same reason as in the autonomouscase.It has become quite clear that there are many nonintegrable systems that pass thesingularity confinement test [20, 21, 3, 33, 16]. Moreover, most linearizable mappings,which are by definition integrable, do not pass the singularity confinement test [32].Besides singularity confinement and algebraic entropy, some other integrability criteriahave also been proposed.Based on Diophantine approximations, Halburd proposed a new integrability criterion,called Diophantine integrability [17]. This approach is particularly useful when we nu-merically estimate the value of the entropy. The coprimeness condition was proposed toreinterpret singularity confinement from an algebraic viewpoint [22, 24, 23]. This criterionfocuses on the factorization of iterates as rational functions of the initial values and triesto transform the equation to another one with the Laurent property [12]. This methodhas been recognized as a useful technique to calculate the exact value of the algebraicentropy [25].Since all equations in this paper are (families of) birational automorphisms, geometricmethods are useful to analyze them. The most important and powerful tool is the so-called space of initial conditions, which was first introduced by Okamoto [30] to analyzethe continuous Painlev´e equations.Sakai focused on the close relation that exists between singularity confinement and aspace of initial conditions. Using a special type of algebraic surface, so-called generalizedHalphen surfaces, he has classified all discrete Painlev´e equations [35].Takenawa performed the blow-ups for the Hietarinta-Viallet equation (regularized asan automorphism on a surface) to obtain its space of initial conditions [36]. He revealeda correspondence between the singularity pattern and the motion of specific curves, andrecalculated the algebraic entropy by computing the maximum eigenvalue of the lineartransformation induced on the Picard group. He also considered blow-ups of nonau-tonomous systems and showed, by using specific bases first introduced by Sakai, that thedegree growth of every discrete Painlev´e equation is at most quadratic [37].Let us start by recalling the close relationship between singularity confinement and thespace of initial conditions. Example 1.5.
Consider the equation(1.2) x n +1 = ( x n + 3 a ) x n − − ax n x n − a , where a is a nonzero constant. Although the gauge x n = a e x n enables us to take a = 1without loss of generality, this gauge does not work on the nonautonomous version of theequation we will consider in Example 2.1. Thus, we do not use the gauge even in theautonomous case.First let us explain the singularity confinement property on the above equation. Let ε be an infinitesimal quantity and assume that while x n − takes a regular finite value, x n becomes 3 a + ε . Then we obtain x n +1 = 6 a ( − a + x n − ) ε − + o ( ε − ) , x n +2 = a + o (1) ,x n +3 = 12 a ( a − x n − ) ε − + o ( ε − ) , x n +4 = − a + o (1) ,x n +5 = 6 a ( − a + x n − ) ε − + o ( ε − ) , x n +6 = − a + o (1) ,x n +7 = − x n − + o (1) , here “ o ( ε k )” is the Landau symbol, i.e. lim ε → o ( ε k ) /ε k = 0. Since the leading order of x n +7 is degree 0 and the leading coefficient again depends on the initial value x n − , wesay that this singularity is confined and its pattern is(1.3) { a, ∞ , a, ∞ , − a, ∞ , − a } . This equation has one more confined singularity pattern(1.4) {− a, − a, a, a } , which starts with x n = − a + ε .Next, we blow up P × P to obtain a space of initial conditions. Equation (1.2) can bewritten as(1.5) ϕ : P × P P × P , ( x, y ) (cid:18) y, ( y + 3 a ) x − ayy − a (cid:19) . Introducing the variables s = 1 /x and t = 1 /y , P × P is covered with 4 copies of C as: P × P = ( x, y ) ∪ ( s, y ) ∪ ( x, t ) ∪ ( s, t ) . Let X be the surface obtained by blowing up P × P at the following 9 points (Figure 1): • P (1) : ( x, t ) = (3 a, • P (2) : ( s, y ) = (0 , a ), • P (3) : ( x, t ) = ( a, • P (4) : ( s, y ) = (0 , − a ), • P (5) : ( x, t ) = ( − a, • P (6) : ( s, y ) = (0 , − a ), • Q (1) : ( x, y ) = ( − a, − a ), • Q (2) : ( x, y ) = ( − a, a ), • Q (3) : ( x, y ) = ( a, a ).Then, ϕ becomes an automorphism on X .Let D (1) , D (2) ⊂ X be the strict transforms of the lines { x = ∞} and { y = ∞} in P × P , respectively, and let C ( i ) , e C ( i ) ⊂ X be the exceptional curves of the blow-up at P ( i ) , Q ( i ) , respectively. Let { y = ± a } , { x = ± a } ⊂ X be the strict transforms of thecorresponding lines in P × P . These curves move under ϕ as follows: D (1) → D (2) → D (1) , { y = 3 a } → C (1) → C (2) → · · · → C (6) → { x = − a } , (1.6) { y = − a } → e C (1) → e C (2) → e C (3) → { x = 3 a } . (1.7)We thus find an exact correspondence between the singularity pattern (1.3) and the mo-tion of curves (1.6) on the one hand, and between the pattern (1.3) and the motion (1.7) onthe other hand, as in P × P these curves correspond to the points P (1) , . . . , P (6) , Q (1) , Q (2) , Q (3) .After sufficient steps, however, these points again become curves. This phenomenon cor-responds to the recovery of the information on the initial value, and thus gives a geometricinterpretation of singularity confinement.We will see in § X . According to Takenawa [36], the maximum eigenvalue of the linearaction gives the dynamical degree of the equation. Using D (1) + C (2) + C (4) + C (6) ∼ { x = − a } + e C (1) ∼ { x = 3 a } + C (1) , = ∞ y = 3 ay = ay = − ay = − ay = 0 x = 0 − a − a a a ∞ t P (1) t P (2) t P (3) t P (4) t P (5) t P (6) t Q (1) t Q (2) t Q (3) Figure 1.
Diagram showing the centers of the blow-ups needed to obtaina space of initial conditions for the mapping (1.5).we have { x = − a } ∼ D (1) + C (2) + C (4) + C (6) − e C (1) , { x = 3 a } ∼ D (1) − C (1) + C (2) + C (4) + C (6) , where “ ∼ ” means the linear equivalence. Thus, the matrix of ϕ ∗ : Pic X → Pic X withrespect to the basis D (1) , D (2) , C (1) , . . . , C (6) , e C (1) , e C (2) , e C (3) is(1.8) −
11 0 1 10 1 0 0 00 1 0 1 10 1 0 0 00 1 1 10 − . In fact, since the eigenvalues of this matrix all have modulus 1, the entropy of this equationis 0.In the above example, we started with P × P and only used blow-ups to obtain a spaceof initial conditions. However, it is also possible to start with P (or a Hirzebruch surface F a ) and, in general, blow-downs are also necessary to obtain a space of initial conditions. f we admit the use of blow-downs, we can take an arbitrary rational surface as a startingpoint. Therefore, the definition of a space of initial conditions is as follows: Definition 1.6 (space of initial conditions for autonomous systems) . If for an autonomousequation ϕ : P P , there exist a rational surface X and a birational map f : X P such that f − ◦ ϕ ◦ f is an automorphism on X : X ∼ / / f (cid:15) (cid:15) X f (cid:15) (cid:15) P ϕ / / P , then X is called a space of initial conditions for ϕ . That is, an autonomous equation hasa space of initial conditions if it can be regularized as an automorphism on some rationalsurface.It is important to note that in general f is a composition of a finite number of blow-upsand blow-downs. Remark 1.7.
Consider an autonomous equation ϕ : P P with a space of initialconditions f : X P and assume that the degree of ϕ n is unbounded. In this case, X has infinitely many exceptional curves of the first kind and thus Theorem A.17 impliesthat there exists a birational morphism g : X → P . Let ψ = g ◦ f − ◦ ϕ ◦ f ◦ g − . Then ψ is a birational automorphism on P : P ψ / / P X ∼ / / f ~ ~ g ` ` ❆❆❆❆❆❆❆❆ X f ! ! g = = ⑤⑤⑤⑤⑤⑤⑤⑤ P ϕ / / P . If we identify two equations that are transformed to each other by a coordinate changeof P , then ϕ and ψ are the same equation. Therefore, by changing coordinates on P appropriately, we can think of f in Definition 1.6 as a composition of blow-ups as long asthe degree growth of the equation is unbounded.All automorphisms on rational surfaces have been classified by Gizatullin [13]. On theother hand, all birational automorphisms on surfaces have been classified by Diller andFavre [8]. Extracting the classification of birational automorphisms on rational surfacesfrom their theorem and interpreting it from the viewpoint of integrable systems, we havethe following classification of autonomous equations of the plane: Theorem 1.8 (Diller-Favre [8]) . Autonomous equations ϕ of the plane are classified intothe following classes: class 1: The degree of ϕ n is bounded.This type of equation has a space of initial conditions.For example, projective transformations on P and periodic mappings belong tothis class. class 2: The degree of ϕ n grows linearly.This type of equation does not have a space of initial conditions.Most linearizable mappings belong this class. lass 3: The degree of ϕ n grows quadratically.This type of equation has a space of initial conditions. It is an elliptic surfaceand ϕ preserves the elliptic fibration on the surface.For example, the QRT mappings belong to this class [31, 39, 11] . class 4: The degree of ϕ n grows exponentially but the equation has a space of initial condi-tions.Its Picard number is greater than .For example, the Hietarinta-Viallet equation belongs to this class. class 5: The degree of ϕ n grows exponentially and the equation does not have a space ofinitial conditions.“Most” equations belong to this class. Moreover, Diller and Favre showed that the value of the dynamical degree of an equationis quite restricted.
Definition 1.9.
A reciprocal quadratic integer is a root of λ − aλ +1 = 0 for some integer a . A real algebraic integer λ > λ > /λ is a conjugate and all(but at least one) of the other conjugates lie on the unit circle. Remark 1.10.
It goes without saying that reciprocal quadratic integers greater than 1and Salem numbers are by definition irrational.
Theorem 1.11 (Diller-Favre [8]) . The dynamical degree of an autonomous equation ofthe plane is , a Pisot number or a Salem number. Theorem 1.12 (Diller-Favre [8]) . If an autonomous equation of the plane has a spaceof initial conditions, then its dynamical degree must be , a reciprocal quadratic integergreater than or a Salem number. If the dynamical degree is , then the degree growthis bounded or quadratic. In particular, this implies that if the degree grows linearly, thenthe equation does not have a space of initial conditions. Theorem 1.12 says that if a mapping has a space of initial conditions, then the valueof its dynamical degree (and algebraic entropy) is strongly restricted. Thus, it is some-times possible to prove the nonexistence of a space of initial conditions by calculating thealgebraic entropy [25].It is well-known that there is a close relation between the degree growth of an equationand the Picard number of its space of initial conditions:
Proposition 1.13.
If an equation has a space of initial conditions with the Picard numberless than (resp. ), then its degree growth is unbounded (resp. at most quadratic).Moreover, if the degree growth is quadratic and a space of initial conditions is minimal(Definition 4.1), then its Picard number is . All autonomous mappings with quadratic degree growth have been classified in [6].Moreover, there is a strong result about equations with bounded degree:
Theorem 1.14 (Blanc-D´eserti [5]) . Let ϕ be a nonperiodic equation with bounded degreegrowth and let X be a space of initial conditions. Then, ϕ can be minimized from X toeither P or a Hirzebruch surface F a with a = 1 . Furthermore, ϕ is birationally conjugateto a projective transformation on P . Therefore, besides periodic mappings, all autonomous integrable (zero algebraic en-tropy) equations of the plane are characterized by a minimal space of initial conditionswith Picard number less than 11. hile the most famous class of nonautonomous equations that have a space of initialconditions is that of the discrete Painlev´e equations, there are many other examples. Forinstance, using algebro-geometric methods, Takenawa considered a nonautonomous exten-sion of the Hietarinta-Viallet equation [36, 37, 38]. In addition, one of the most importantand powerful methods to find a nonautonomous equation with all singularities confined isso-called late confinement, which was first reported in [21]. This method provides us witha family of nonautonomous equations that pass the singularity confinement test [28, 16].Until now, no general theory of nonautonomous mappings with a space of initial con-ditions has been formulated. One of the main aims of this paper is such a classificationof integrable equations with a space of initial conditions. It is known that all discretePainlev´e equations have a space of initial conditions (by definition [35]) and that theyare integrable (as shown by Takenawa [37]). Then, is it conceivable that there existsan integrable equation that is not a discrete Painlev´e equation but which has a space ofinitial conditions?The reason why there has been almost no general theory of spaces of initial conditionsin the nonautonomous case is the difficulty in setting up a suitable starting point. Inthe autonomous case, an equation with a space of initial conditions is reduced to oneautomorphism on a single rational surface. However, even if a nonautonomous systemsuch as a discrete Painlev´e equation has a space of initial conditions, it is in general notreducible to an automorphism on a surface. Furthermore, in the nonautonomous case,even the centers of the blow-ups and therefore the obtained surface do depend on n . Asa result, a space of initial conditions is not a single surface in a strict sense, but rather afamily of surfaces. Therefore, choosing appropriate ϕ n , one can obtain many pathologicalexamples. It is true that this kind of problem does not matter when we consider aconcrete example such as a discrete Painlev´e equation or a nonautonomous extensionof the Hietarinta-Viallet equation. However, if we are interested in a classification, wecannot avoid the need to set up an appropriate starting point. In §
2, we shall first describeseveral artificial examples and then define a space of initial conditions for nonautonomousequations. We will also recall the space of initial conditions in Sakai’s sense and showthat these two definitions are equivalent. § §
4, we shall see that a minimization of a space of initial conditions in the nonautonomouscase is in fact quite similar to that in the autonomous case. § § or the convenience of the reader, in Appendix A, we describe the notations we usethroughout the paper and recall some basic results on algebraic surfaces. Appendix B is anelementary but rather involved proof of a fundamental fact in linear algebra (Lemma 3.7).2. Space of initial conditions for nonautonomous systems
In this section, we define a space of initial conditions in the nonautonomous case. Firstwe show several examples in order to explain what is necessary in the definition of a spaceof initial conditions. Next, we will state the definition (Definition 2.5). Finally, we willsee in Proposition 2.21 that there is a correspondence between our definition of a spaceof initial conditions and that of Sakai.
Example 2.1.
Let us consider a nonautonomous extension of the equation in Exam-ple 1.5: y n +1 = ( y n + 3 a n − α ) y n − − a n y n − αa n y n − a n − α , where α is a constant and a n satisfies a n +1 = a n + α = 0 [34]. We can recover theautonomous case by taking α = 0. As in the autonomous case, this equation has twoconfined singularity patterns: { a n + 2 α, ∞ , a n , ∞ , − a n − α, ∞ , − a n − α } and {− a n + 2 α, − a n + 3 α, a n + 6 α, a n + 11 α } . Let us regularize this equation as a family of isomorphisms on surfaces by blow-ups.The equation can be written as follows: ϕ n : P × P P × P , ( x n , y n ) (cid:18) y n , ( y n + 3 a n − α ) x n − a n y n − αa n y n − a n − α (cid:19) . Using the variables s n = 1 /x n and t n = 1 /y n , we cover P × P with 4 copies of C : P × P = ( x n , y n ) ∪ ( s n , y n ) ∪ ( x n , t n ) ∪ ( s n , t n ) . Let X n be the surface obtained by blowing up P × P at the following 9 points: • P (1) n : ( x n , t n ) = (3 a n − α, • P (2) n : ( s n , y n ) = (0 , a n − α ), • P (3) n : ( x n , t n ) = ( a n − α, • P (4) n : ( s n , y n ) = (0 , − a n − α ), • P (5) n : ( x n , t n ) = ( − a n − α, • P (6) n : ( s n , y n ) = (0 , − a n + 2 α ), • Q (1) n : ( x n , y n ) = ( − a n + 5 α, − a n + 4 α ), • Q (2) n : ( x n , y n ) = ( − a n + 5 α, a n + 4 α ), • Q (3) n : ( x n , y n ) = ( a n + 3 α, a n + 2 α ).The configuration of these 9 points is almost the same as in the autonomous case (Fig-ure 1). Since ϕ n (cid:0) P ( i ) n (cid:1) = P ( i +1) n +1 for i = 1 , . . . , ϕ n (cid:0) Q ( i ) n (cid:1) = Q ( i +1) n +1 for i = 1 ,
2, one finds that ϕ n is indeed an isomorphism from X n to X n +1 .As in the autonomous case, let us label specific curves on X n as follows: D (1) n , D (2) n : the strict transforms of the lines { x = ∞} and { y = ∞} in P × P ,respectively. • C ( i ) n : the exceptional curve of the blow-up at P ( i ) n . • e C ( i ) n : the exceptional curve of the blow-up at Q ( i ) n .These curves move under the equation as: D (1) → D (2) → D (1) , { y = 3 a + 2 α } → C (1) → C (2) → · · · → C (6) → { x = − a + 5 α } , { y = − a + 2 α } → e C (1) → e C (2) → e C (3) → { x = 3 a − α } , where we omit the index n . Thus, the matrix of ϕ ∗ : Pic X n → Pic X n +1 with respect tothe basis ( D (1) , D (2) , C (1) , . . . , C (6) , e C (1) , e C (2) , e C (3) ) coincides exactly with the one obtainedin the autonomous case (1.8). Therefore, the algebraic entropy of this equation is zero aswell.As we have seen in the above example, when considering a space of initial conditions,it is most important for the equation to be regularized as a (family of) isomorphism(s) onsurfaces. However, since there exist lots of pathological nonautonomous equations, thiscondition is so weak that we cannot hope to say anything about general properties of suchequations.Let us consider some of these examples. In the following examples, we fix the startingindex at n = 0, i.e. by deg ψ n we denote deg( ψ n − ◦ · · · ◦ ψ ) (Remark 1.2). Example 2.2.
Let ϕ be an arbitrary autonomous equation with unbounded degree growthand a space of initial condition X (for example, the mapping ϕ in Example 1.5), and let( d n ) n> be an arbitrary sequence of positive integers. Define sequences ( p n ) n ≥ and ( q n ) n> by p = 0 , p n = max { k ∈ Z ≥ | deg ϕ k ≤ d n } , q n = p n − p n − . Let ψ n = ϕ q n : P P for all n >
0. Then, we havedeg( ψ n ◦ · · · ◦ ψ ) = deg ϕ p n ≈ d n . Since ϕ is an automorphism on X , so is ψ n for all n . Therefore, by choosing ( d n ) n appro-priately, we can construct many equations that can be reduced to families of isomorphisms(automorphisms) on surfaces but that have arbitrary degree growth.Case 1Let λ be an arbitrary real number greater than 1 and let d n be the greatest integer notgreater than λ n . In this case, the entropy of the mapping ψ n is log λ .Case 2Let λ as in Case 1 and let d n = ( the greatest integer not greater than λ n ( n : even)1 ( n : odd) . In this case, the entropy of the mapping ψ n does not exist. If we change the definition ofthe entropy to lim sup n →∞ n log (deg ψ n ) , then the entropy exists and is log λ . ase 3Let d n = n . In this case, the degree of ψ n grows linearly but the equation can bereduced to a family of automorphisms on X (in contrast to Theorem 1.8, class 2 forautonomous mappings).Case 4Let d n grow faster than any exponential function of n , for example d n = n n . In thiscase, the entropy of the mapping ψ n is + ∞ (in contrast to Remark 1.2 for autonomousmappings). Example 2.3.
The same technique as above can also be used in case the original ϕ doesnot have a space of initial conditions. Let ϕ be an autonomous equation with unboundeddegree growth but no space of initial conditions (for example a linearizable mapping) andlet d n = n . Then, we obtain a mapping ψ n that has a quadratic degree growth butcannot be regularized as a family of isomorphisms on surfaces. Example 2.4.
In the above two examples, the equations are quite artificial and practicallyimpossible to write explicitly. Usually, the term “nonautonomous equation” refers to anequation with several nonautonomous coefficients such as Example 2.1. However, even inthis class of equations, there are strange mappings.Consider the equation x n +1 = a n x n + (1 − a n ) x n + bx n − , where b is a general constant and a n is a nonautonomous coefficient. We are interestedonly in the case a n = 0 , a n is always 0, this equation is a linear mapping and thus the degreegrowth is obviously bounded. On the other hand, in the case where a n is always 1, thisequation is a H´enon map [19] and its algebraic entropy is log 2.If a n can take both values 0 and 1, then these two cases are mixed. It is obvious that forany real number λ ∈ [1 , a n ) n such that the dynamical degreeof the above equation is λ .It is always possible to mix two different equations by using one nonautonomous coeffi-cient. For example, if we start with two autonomous equations that have the same spaceof initial conditions, then the mixed equation is reduced to a family of automorphisms ona surface but exhibits strange behavior.What is important is that, even if the obtained surfaces and isomorphisms depend on n , their “fundamental structures” (for example, the intersection pattern of specific curvesand the linear action induced on the Picard groups) are the same. When we consider aconcrete equation such as Example 2.1, it is (in principle) possible to check whether thosestructures do or do not depend on n . However, it is difficult to define mathematicallywhat constitutes a fundamental structure for general equations. In this paper we shalltherefore define a space of initial conditions as follows: Definition 2.5 (space of initial conditions for nonautonomous systems) . An equation ϕ n : P P has a space of initial conditions if (after an appropriate coordinate change)the following three conditions are satisfied: There exists a composition of blow-ups π n = π (1) n ◦ · · · ◦ π ( r ) n : X n → P for each n such that the induced birational maps ϕ n : X n X n +1 are all isomorphisms: / / X n − π n − (cid:15) (cid:15) ∼ / / X nπ n (cid:15) (cid:15) ∼ / / X n +1 π n +1 (cid:15) (cid:15) / / / / P ϕ n − / / P ϕ n / / P / / . • Let e n = ( e (0) n , . . . , e ( r ) n ) be the geometric basis corresponding to π n (Definition A.8).Then, the matrices of ϕ n ∗ : Pic X n → Pic X n +1 with respect to these bases do notdepend on n . • The set of all effective classes in Pic X n does not depend on n , i.e. if P i a ( i ) e ( i ) n ∈ Pic X n is effective, then so is P i a ( i ) e ( i ) k ∈ Pic X k for any k .We refer the reader to Appendix A for an explanation of the notations and for asummary of some basic results on algebraic surfaces.Note that in the nonautonomous case, a space of initial conditions does not consist ofa single surface but of a family of surfaces. It also contains information about the centersand ordering of the blow-ups. Remark 2.6.
As in the autonomous case (Remark 1.7), blow-downs are necessary, ingeneral, to construct a space of initial conditions. However, to avoid unnecessary com-plexity, we use the phrase “after an appropriate coordinate change” instead. We will seein Remark 2.22 the rigorous definition including blow-downs.Usual nonconfining equations such as linearizable mappings and H´enon maps do notsatisfy the first condition in Definition 2.5. On the other hand, Example 2.2 does satisfythe first and third conditions but does not satisfy the second.The third condition imposes some constraint on the centers and ordering of blow-ups.Unfortunately, it is not easy in general to check the third condition in Definition 2.5 for aconcrete equation. However, we shall see that even if only the first and second conditionsare satisfied, we can still calculate the degree growth by Proposition 3.2 (since its proofdoes not need the third condition). One reason why we introduce the third condition is thecorrespondence to a space of initial conditions in Sakai’s sense, which we shall introducelater.
Remark 2.7.
Let us first have a closer look at the second condition. Let Z ,r = Z e (0) ⊕ · · · ⊕ Z e ( r ) and define on Z ,r a symmetric bilinear form ( − , − ) by( e ( i ) , e ( j ) ) = i = j = 0) − i = j = 0)0 ( i = j ) . Let ι n : Z ,r → Pic X n , e ( i ) e ( i ) n and Φ n = ι − n +1 ϕ n ∗ ι n : / / Z ,r Φ n − / / ι n − (cid:15) (cid:15) Z ,r Φ n / / ι n (cid:15) (cid:15) Z ,r / / ι n +1 (cid:15) (cid:15) / / PicX n-1 ϕ n − ∗ / / PicX n ϕ n ∗ / / PicX n+1 / / . hen, the meaning of the second condition is that Φ n does not depend on n . We thensimply denote Φ n by Φ.We will use these notations in § Lemma 2.8.
Let K = ι − n K X n = − e (0) + e (1) + · · · + e ( r ) . Then Φ preserves K and ( − , − ) , i.e. Φ K = K, ( v, w ) = (Φ v, Φ w ) for all v, w ∈ Z ,r .Proof. Immediate from the fact that ϕ n ∗ preserves the canonical class and the intersectionnumber on the surface. (cid:3) Next, we review the notion of a space of initial conditions in Sakai’s sense.Let X be a basic rational surface (Definition A.7). Let e = ( e (0) , . . . , e ( r ) ) , e e = ( e e (0) , . . . , e e ( r ) )be geometric bases and π, e π : X → P the corresponding birational morphisms. Then weobtain a birational automorphism e π ◦ π − : P P which will become a part of anequation.Let σ be the Z -linear map on Pic X defined by e (0) e e (0) , . . . , e ( r ) e e ( r ) . Suppose that σ n e = ( σ n e (0) , . . . , σ n e ( r ) ) is a geometric basis for each n and let π n : X → P be the corresponding birational morphism. Then, we obtain the equation ϕ n = π n +1 ◦ π − n : P P . Example 2.9.
The following is probably the simplest example where σ n e is not a geo-metric basis.We cover P by three copies of C as follows: P = ( x, y ) ∪ (cid:18) xy , y (cid:19) ∪ (cid:18) yx , x (cid:19) . Let π (1) , π (2) , π (3) be the blow-ups at the following points: • π (1) : at P (1) : ( x, y ) = (0 , • π (2) : at P (2) : (cid:18) y , xy (cid:19) = (0 , • π (3) : at P (3) : (cid:18) x , xy (cid:19) = (0 , X be the surface obtained by the blow-ups π = π (1) ◦ π (2) ◦ π (3) (Figure 2) and let e = ( e (0) , e (1) , e (2) , e (3) ) be the corresponding geometric basis.It is obvious that e e = ( e e (0) , e e (1) , e e (2) , e e (3) ) = ( e (0) , e (2) , e (3) , e (1) )is another geometric basis on Pic X . Let σ be the Z -linear transformation on Pic X definedby e ( i ) e e ( i ) for all i . While e and σe = e e are geometric, σ e = ( e (0) , e (3) , e (1) , e (2) ) is notsince e (2) − e (3) is effective.It is obvious that all problems in this case come from the ordering of the e ( i ) .As in the above example, σ n e is not always geometric. Therefore, it is necessary toimpose some condition on σ . Definition 2.10 (Cremona isometry [27, 9, 35]) . Let X be a rational surface and let σ be an invertible Z -linear transformation on Pic X . σ is said to be a Cremona isometry ifit satisfies the following three conditions: ❅❅❅❅❅❅❅❅❅ t P (1) t P (2) → x ↑ y ← /x տ y/x ↓ /y ց x/y ↓ ❅❅❅❅❅ ❅❅❅✟✟✟✟ t P (3) ↓ ❅❅❅ ❅❅❅PPPPP Figure 2.
Diagram showing the blow-ups needed to obtain X in Example 2.9. • σ preserves the intersection number on Pic X , i.e. F · F = ( σF ) · ( σF ) for all F , F ∈ Pic X , • σ preserves K X , • σ preserves the set of effective classes, i.e. if F is effective, then so is σF (and σ − F ). Example 2.11.
Let ϕ be an automorphism on a rational surface. Then the inducedlinear transformations ϕ ∗ and ϕ ∗ are Cremona isometries.It is clear from the definition that the following holds. Lemma 2.12.
Cremona isometries preserve the nef cone.
It should be noted that, while an automorphism on a surface determines the motion ofeach curve, a Cremona isometry does not . It only determines the motion of the classes ofcurves. However, as shown in the following lemma, if an irreducible curve has a negativeself-intersection, then its motion is completely determined.
Lemma 2.13.
Let X be a rational surface and σ a Cremona isometry, and let C be anirreducible curve in X with negative self-intersection. Then there exists only one effective ivisor D such that [ D ] = σ [ C ] . Moreover, D is a prime divisor, i.e. an irreducible curve.In particular, σ acts as a permutation on the set of all exceptional curves of the first kind.Proof. Let σ [ C ] = " k X i =1 m i C i , where C i are irreducible curves. Since[ C ] = k X i =1 m i σ − [ C i ]and σ − [ C i ] are all effective, it follows from Proposition A.9 that k = 1 and m = 1. (cid:3) Lemma 2.14.
Let σ be a Cremona isometry. If e = ( e (0) , . . . , e ( r ) ) is a geometric basison Pic X , then so is σe = ( σe (0) , . . . , σe ( r ) ) .Proof. Let π = π (1) ◦ · · · ◦ π ( r ) : X → P be the composition of blow-ups correspondingto e and let C , . . . , C r ⊂ X be the irreducible curves contracted by π . Since all thesecurves have negative self-intersection, by Lemma 2.13, their motions are determined by σ . Let us denote them by C ′ , . . . , C ′ r . Since C i · C j = C ′ i · C ′ j for all i, j , it is possible tocontract C ′ , . . . , C ′ r in the same order as C , . . . , C r . It is clear that the geometric basiscorresponding to this contraction is σe . (cid:3) Let us see how to obtain an equation from a Cremona isometry [35].
Definition 2.15.
Let X be a basic rational surface and let σ be a Cremona isometry onPic X and take an arbitrary geometric basis e = ( e (0) , . . . , e ( r ) ). By Lemma 2.14, σ n e is ageometric basis for each n . Let π n be the corresponding birational morphism to P andlet ϕ n = π n +1 ◦ π − n . Thus we obtain ( ϕ n ) n ∈ Z , a family of birational automorphisms on P : X π n − ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ π n (cid:15) (cid:15) π n +1 ❆❆❆❆❆❆❆❆ / / P ϕ n − / / P ϕ n / / P / / . This is the equation defined by
X, σ and e , and we call X a space of initial conditions(in Sakai’s sense). Since the choice of e only determines the specific coordinates, wesometimes think of ( X, σ ) as the equation itself.Note that ( ϕ n ) n ∈ Z is determined by X, σ and e up to an automorphism of P for each n , i.e. if ( ϕ ′ n ) n ∈ Z is another family of birational automorphisms on P defined by X, σ and e , then there exist f n ∈ PGL(3 , C ) such that ϕ n = f n +1 ◦ ϕ ′ n ◦ f − n . Definition 2.16 (generalized Halphen surface [35]) . A rational surface X is called a generalized Halphen surface if it satisfies the following two conditions: • − K X is effective, • All components of − K X are orthogonal to − K X , i.e. D i · ( − K X ) = 0 for any P i m i D i ∈ | − K X | . Lemma 2.17 (Proposition 2 in [35]) . Any generalized Halphen surface is a basic rationalsurface. efinition 2.18 (discrete Painlev´e equation [35]) . Let X be a generalized Halphen surfaceand σ a Cremona isometry on Pic X of infinite order. Then, the equation obtained bythe above procedure is called a discrete Painlev´e equation . Remark 2.19.
Note that according to this definition, autonomous mappings such as theQRT mappings are also labeled “discrete Painlev´e.”Using generalized Halphen surfaces, Sakai has classified (and, in a sense, defined) alldiscrete Painlev´e equations. Since we do not need such a detailed classification in thispaper, we only give a brief summary.The orthogonal lattice K ⊥ X ⊂ Pic X , which is preserved under σ , is an affine root latticeof type E (1)8 . If dim | − K X | = 0, then the expression P i m i D i ∈ | − K X | is unique.Therefore, σ acts on the set { D i } i as a permutation and preserves the lattice span Z D i and its orthogonal compliment. These two lattices are both affine root sublattices of K ⊥ X and play an important role in the classification of the discrete Painlev´e equations. Remark 2.20.
While Cremona isometries can be defined for any rational surface, weonly consider basic rational surfaces such as in Definition 2.15. Although it is possible toconsider a family of blow-downs from a nonbasic rational surface to a Hirzebruch surfaceF a ( a ≥
2) instead of P , Theorem A.17 implies that the degree growth of such an equationmust be bounded. Hence, it is sufficient to consider only basic rational surfaces as longas we are interested in equations with unbounded degree growth.Now let us clarify the correspondence between the two definitions of a space of initialconditions we considered. Proposition 2.21.
The two definitions of a space of initial conditions, Definition 2.5and Definition 2.15, are equivalent.Proof.
First, consider the situation in Definition 2.5. Let X = X and σ be the Z -lineartransformation on Pic X defined by σ = ι Φ − ι − . Then, a direct calculation shows that σ ℓ e ( i )0 = ϕ ∗ · · · ϕ ∗ ℓ − e ( i ) ℓ , σ − ℓ e ( i )0 = ϕ − ∗ · · · ϕ − ℓ ∗ e ( i ) − ℓ for all ℓ > σ is a Cremona isometry. It is clear, by construction, that σ satisfiesthe first and second conditions on a Cremona isometry. Let F = P i a ( i ) e ( i )0 ∈ Pic X bean effective class. Then we have σF = X i a ( i ) ϕ ∗ e ( i )1 = ϕ ∗ X i a ( i ) e ( i )1 ! . The third condition in Definition 2.5 implies that P i a ( i ) e ( i )1 is effective. Since ϕ ∗ preservesthe set of effective classes, σF is also effective. We can prove the effectiveness of σ − F inthe same way.Next, consider the situation in Definition 2.15. That is, X is a basic rational surfaceand σ is a Cremona isometry on Pic X . Take e = ( e (0) , . . . , e ( r ) ) as a geometric basis onPic X and consider the equation defined by X, σ and e . Let us recover the above situationfrom these data. et X n = X and e n = σ n e for all n ∈ Z . While the X n themselves are all the same,the bases e n vary depending on n . Then we have the following diagram: X n − X n X n +1 / / X π n − (cid:15) (cid:15) id / / X π n (cid:15) (cid:15) id / / X π n +1 (cid:15) (cid:15) / / / / P ϕ n − / / P ϕ n / / P / / . It is important to note that, while the morphisms from X n to X n +1 are all the identitymap on X n = X n +1 = X , the ϕ n are not the identity map on P in general.Let us check the second condition in Definition 2.5. Let A n = (cid:16) a ( i,j ) n (cid:17) i,j be the matrixrepresentation of ϕ n ∗ with respect to the bases e n and e n +1 . Since ϕ n ∗ = id Pic X , A n aredetermined by e ( i ) n = X j a ( j,i ) n e ( j ) n +1 . Applying σ k and using σ k e ( i ) n = e ( i ) n + k , we have e ( i ) n + k = X j a ( j,i ) n e ( j ) n + k +1 , which shows that A n do not depend on n .Finally we check the third condition in Definition 2.5. Let F = P i a ( i ) e ( i ) n ∈ Pic X bean effective class. Since σ k preserves the effective classes, σ k F = X i a ( i ) e ( i ) n + k are effective for all k . Hence the set of effective classes does not depend on n . (cid:3) We have seen that the two definitions of a space of initial conditions are equivalent. Inthis paper, we will use both definitions depending on the situation.
Remark 2.22.
If we consider blow-downs instead of an “appropriate coordinate change”in Definition 2.5, we must assume that the blow-downs do not depend on n . In this case,one possible rigorous definition is as follows:An equation ( ϕ n ) n has a space of initial conditions if there exist rational surfaces Y n and X n , blow-ups π n = π (1) n ◦ · · · ◦ π ( r ) n : Y n → P and blow-downs ǫ n = ǫ (1) n ◦ · · · ◦ ǫ ( r ′ ) n : Y n → X n for each n , such that the following four conditions are satisfied: • ϕ n is an isomorphism from X n to X n +1 . • Let e e n = ( e e (0) n , . . . , e e ( r ) n ) be the geometric basis corresponding to π n and identifyall Pic Y n by these bases. Let E ( k ) n be the total transform of the exceptional classof ǫ ( k ) n . Then, E ( k ) n does not depend n . • Take a basis e n = ( e (0) n , . . . , e ( r − r ′ +1) n ) of Pic X n for each n such that ǫ ∗ n e ( i ) n does notdepend on n (under the above identification). Identify all Pic X n by these bases.Then, ϕ n ∗ does not depend on n . • The set of effective classes in Pic X n (and in Pic Y n ) does not depend on n (underthe above identification). n − (cid:15) (cid:15) ●●●●●●●●● Y n (cid:15) (cid:15) ❆❆❆❆❆❆❆❆ Y n +1 (cid:15) (cid:15) ●●●●●●●●● / / X n − (cid:15) (cid:15) / / X n (cid:15) (cid:15) / / X n +1 (cid:15) (cid:15) / / / / P / / P / / P / / / / P / / P / / P / / Figure 3.
Diagram showing a space of initial conditions in the case wherewe consider blow-downs. The second row from the bottom represents theoriginal equation and the bottom row represents a new equation obtainedby an appropriate coordinate change.As in the autonomous case (Remark 1.7), if the equation has unbounded degree growth,then it is possible to reduce the above situation to that in Definition 2.5 by taking newblow-downs X n → P (Figure 3). Needless to say, the new blow-downs must be suchthat the geometric basis on Pic X n does not depend on n . As in the autonomous case,the existence of such blow-downs is guaranteed by Theorem A.17. Hence, as long as weare interested only in performing a classification, we may only consider the situation inDefinition 2.5.The reason why this kind of problem arises is that we start from a specific equation( ϕ n ) n , whereas if we start from the situation in Definition 2.15, this kind of problem doesnot appear.From now on, we shall assume that a space of initial conditions is obtained only byblow-ups, i.e. we shall simply consider the situation in Definition 2.5 or Definition 2.15.3. Basic properties of an equation with a space of initial conditions
In this section, we first recall Takenawa’s result on the degree growth for an equation[37]. Next we shall see that, as in the autonomous case, the degree growth of a nonau-tonomous equation with a space of initial conditions can be classified into three cases.Finally we show some relations between the degree growth of an equation and the Picardnumber of a space of initial conditions.In this section, we consider the situation in Definition 2.5. We will also use the Φ and ι n defined in Remark 2.7.Since we will not use the third condition in Definition 2.5 in this section, the resultswill still hold in the case where the third condition is not satisfied. Lemma 3.1 (Takenawa [37]) . deg ϕ n = (Φ n e (0) , e (0) ) . roof. Consider the following commutative diagram: Z ,rι ℓ (cid:15) (cid:15) Φ / / · · · Φ / / Z ,rι ℓ + n (cid:15) (cid:15) Pic X ℓ ϕ ℓ ∗ / / · · · ϕ ℓ + n − ∗ / / Pic X ℓ + nπ ℓ + n ∗ (cid:15) (cid:15) Pic( P ) π ∗ ℓ O O Pic( P ) . Using Definition A.6, we havedeg ϕ n = ( π ℓ + n ∗ ( ϕ ℓ + n − ◦ · · · ◦ ϕ ℓ ) ∗ π ∗ ℓ O P (1)) · O P (1)= (cid:16) ( ϕ ℓ + n − ◦ · · · ◦ ϕ ℓ ) ∗ e (0) ℓ (cid:17) · e (0) ℓ + n = (cid:0) ι ℓ + n − Φ n e (0) (cid:1) · e (0) ℓ + n = (Φ n e (0) , e (0) ) . (cid:3) Proposition 3.2.
The Jordan normal form of Φ is one of the following three: • Φ ∼ µ . . . µ r +1 , where µ i are all roots of unity. In particular, there exists ℓ > such that Φ ℓ = id and thus the degree growth of the equation is bounded. • Φ ∼ µ . . . µ r − , where µ i are all roots of unity. In this case, the degree grows quadratically. Thedominant eigenvector is isotropic. • Φ ∼ λ λ µ . . . µ r − , where λ is a reciprocal quadratic integer greater than or a Salem number, and | µ i | = 1 . In this case, the entropy of the equation is log λ > . The two eigenvectorscorresponding to λ and /λ are both isotropic. These three cases correspond to the classes 1 , Corollary 3.3 (Takenawa [37]) . The dynamical degree of an equation is given by themaximum eigenvalue of Φ and the entropy by its logarithm. orollary 3.4. Theorem 1.12 still holds in the nonautonomous case.
Remark 3.5.
We have already seen in Example 2.3 that Theorem 1.11 does not hold ingeneral nonautonomous cases. To extend Theorem 1.11 to the nonautonomous case, it isnecessary to apply some conditions on the mapping ϕ n itself. However, since there existtoo many possible artificial equations in the nonautonomous case, it would be extremelydifficult to describe such conditions in all generality.It is easy to prove Proposition 3.2 if we admit the following two lemmas in linearalgebra. Lemma 3.6.
Let V be an ( r + 1) -dimensional C -vector space with a Hermitian form ( − , − ) of signature (1 , r ) . If v ∈ V is isotropic, i.e. ( v, v ) = 0 and v = 0 , then thesignature of ( − , − ) (cid:12)(cid:12) v ⊥ is (0 , r − and its kernel is generated by v . In particular, if v , v satisfy ( v , v ) = ( v , v ) = ( v , v ) = 0 , then v and v are linearly dependent. Lemma 3.7.
Let V be an ( r + 1) -dimensional R -vector space with a symmetric bilinearform ( − , − ) of signature (1 , r ) , and let f be a linear transformation on V which preserves ( − , − ) . (1) The Jordan normal form of f must be one of the following: (3.1) µ . . . µ r +1 ( | µ i | = 1) , (3.2) ν ν ν µ . . . µ r − ( ν = ± , | µ i | = 1) , (3.3) λ λ µ . . . µ r − ( λ ∈ R , | λ | > , | µ i | = 1) . (2) Consider the case where the Jordan normal form of f is (3.2). If ( v , v , v , u , . . . , u r − ) is the corresponding Jordan basis on V C = V ⊗ C , then v is isotropic and lim n → + ∞ ν n n f n w = ( w, v )2( v , v ) v for any w ∈ V C . (3) Consider the case where the Jordan normal form of f is (3.3). If ( v , v , u , . . . , u r − ) is the corresponding Jordan basis, then v and v are both isotropic and lim n → + ∞ λ n f n w = ( w, v )( v , v ) v for any w ∈ V C . lthough we shall use Lemma 3.6 throughout the paper, we omit its proof since it is awell-known fact in linear algebra. The proof of Lemma 3.7 will be given in Appendix Bsince it is long and not often stated explicitly in the literature. Proof of Proposition 3.2.
By Lemma 3.7, the Jordan normal form of Φ is (3.1), (3.2) or(3.3).Case: (3.1)It is sufficient to show that every eigenvalue of Φ is a root of unity. Since Φ preservesthe lattice Z ,r , its characteristic polynomial has integer coefficients. Since all roots ofthis polynomial have modulus 1, they are all roots of unity by Kronecker’s theorem [26].Case: (3.2)It is clear that the degree growth is at most quadratic, and the reason why the µ i areroots of unity is the same as above. Therefore, it is sufficient to show that ν = 1 and thatthe degree growth is actually quadratic.Using Lemma 3.1 and Lemma 3.7 (2), we havelim n → + ∞ deg ϕ n ν n n = (cid:18) lim n → + ∞ ν n n Φ n e (0) , e (0) (cid:19) = (cid:18) ( e (0) , v )2( v , v ) v , e (0) (cid:19) = ( e (0) , v ) v , v ) . Since v is isotropic, ( e (0) , v ) is not 0 and thus deg ϕ n /ν n grows quadratically. Sincedeg ϕ n is always positive, we have ν = 1.Case: (3.3)Since λ has modulus greater than 1, as in the case of (3.2), we havelim n → + ∞ deg ϕ n λ n = ( e (0) , v )( e (0) , v )( v , v ) . We can prove ( e (0) , v ) = 0, ( e (0) , v ) = 0 and λ > (cid:3) The following proposition shows the relation between the Jordan normal form of Φ andthe Picard number ρ ( X n ). Proposition 3.8. (1) If ρ ( X n ) < , then the degree growth of the equation is bounded. (2) If ρ ( X n ) ≤ , then the degree growth of the equation is bounded or quadratic.Proof. The key to the proof is that Φ preserves K = 3 e (0) − e (1) − · · · − e ( r ) .(1) Since ( K, K ) = K X n = 10 − ρ ( X n ) > , the bilinear form ( − , − ) is negative definite on K ⊥ . Since Φ (cid:12)(cid:12) K ⊥ preserves the lattice K ⊥ with a negative definite bilinear form, there exists ℓ > (cid:0) Φ (cid:12)(cid:12) K ⊥ (cid:1) ℓ = id.Note that the above also implies that K (and − K ) is isotropic if and only if the Picardnumber of X n is 10.(2) Let us assume that the dynamical degree λ is greater than 1 and show that ( K, K ) <
0. Let v ∈ R ,r be the eigenvector corresponding to λ . Since( v, K ) = ( λv, K ) = λ ( v, K ) , e have K ∈ v ⊥ . Lemma 3.6 says that ( − , − ) (cid:12)(cid:12) v ⊥ is semi-negative definite and its kernelis generated by v . Since v and K are eigenvectors corresponding to different eigenvalues,we have K / ∈ C v , and thus ( K, K ) < (cid:3) Since all generalized Halphen surfaces have Picard number 10, one immediately obtainsthe following corollary, which was first shown by Takenawa on a case-by-case basis [37].
Corollary 3.9.
The degree growth of any discrete Painlev´e equation is quadratic. Inparticular, all discrete Painlev´e equations are integrable.
As shown in the following example, the direct converse of Proposition 3.8 does not hold,even in the autonomous case.
Example 3.10.
Let ϕ be an automorphism on a rational surface X and let P ∈ X be a fixed point of ϕ . Let ǫ : e X → X be the blow-up at P . Then, ϕ is lifted to anautomorphism on e X .This procedure does not change the algebraic entropy of an equation but increases thePicard number of a space of initial conditions.When we consider the classification of equations with a space of initial conditions, itis sometimes necessary to perform a minimization of the space. A concrete approach tosuch minimizations was considered by Carstea and Takenawa in [7], where they gave anexample of a minimization that contracts a curve passing through C (the finite regionin P or P × P ). However, a general theory in the nonautonomous case was not yetknown. In the following section, we will consider a minimization of a space of initialconditions in the general case, in order to classify all nonautonomous integrable equationswith unbounded degree growth that possess a space of initial conditions.4. Minimization of a space of initial conditions
Let us consider a minimization of a space of initial conditions for a nonautonomousmapping. In this section, we consider the situation in Definition 2.15 and think of (
X, σ )as defining an equation.We first recall the process of minimization in the autonomous case.
Definition 4.1.
Let ϕ be an autonomous equation (automorphism) on a rational surface X . Then, ϕ is minimal on X if there is no rational surface X ′ , no automorphism ϕ ′ (on X ′ ) and no birational morphism ǫ : X → X ′ , such that ρ ( X ) > ρ ( X ′ ) and ǫ ◦ ϕ = ϕ ′ ◦ ǫ : X ∼ ϕ / / ǫ (cid:15) (cid:15) X ǫ (cid:15) (cid:15) X ′ ∼ ϕ ′ / / X ′ . It is known that an automorphism ϕ on X is minimal if and only if there are no mutuallydisjoint exceptional curves of the first kind on X that are permuted by ϕ . The proof ofthis statement is almost exactly the same as that of Lemma 4.3.Hence, what we call a minimization of an autonomous equation is first of all the processof finding such contractible curves and then to actually realize the contraction. Definition 4.2.
Let X be a rational surface and let σ be a Cremona isometry on X . Anonautonomous equation ( X, σ ) is minimal if there is no rational surface X ′ , no birational orphism ǫ : X → X ′ and no Cremona isometry σ ′ on Pic X ′ , such that ρ ( X ) > ρ ( X ′ )and ǫ ∗ σ = σ ′ ǫ ∗ : Pic X σ / / ǫ ∗ (cid:15) (cid:15) Pic X ǫ ∗ (cid:15) (cid:15) Pic X ′ σ ′ / / Pic X ′ . As in the autonomous case, it is possible to explicitly verify the minimality with specifictypes of curves.
Lemma 4.3.
Let X be a rational surface and σ a Cremona isometry on Pic X . Theequation ( X, σ ) is minimal if and only if there are no mutually disjoint exceptional curvesof the first kind C , . . . , C N ⊂ X that are permuted by σ . Lemma 4.4.
Let
X, X ′ be rational surfaces and ǫ : X → X ′ a birational morphism. If aCremona isometry σ on Pic X preserves the sublattice ǫ ∗ (Pic X ′ ) ⊂ Pic X , then ǫ ∗ σǫ ∗ isalso a Cremona isometry on Pic X ′ .Proof. Let ǫ = ǫ (1) ◦ · · · ◦ ǫ ( L ) be a decomposition into blow-ups and let E ( i ) be the totaltransform of the class of the exceptional curve of ǫ ( i ) for i = 1 , . . . , L .Let F, F ′ ∈ Pic X ′ . Using σǫ ∗ F, σǫ ∗ F ′ ∈ ǫ ∗ (Pic X ′ ), we have( ǫ ∗ σǫ ∗ F ) · ( ǫ ∗ σǫ ∗ F ′ ) = ( σǫ ∗ F ) · ( σǫ ∗ F ′ )= F · F ′ . Since K X = ǫ ∗ K X ′ + E (1) + · · · E ( L ) , ǫ ∗ E ( i ) = 0 and since the E ( i ) are permuted by σ we have that ǫ ∗ σǫ ∗ K X ′ = ǫ ∗ σ ( K X − E (1) − · · · E ( L ) )= ǫ ∗ ( K X − E (1) − · · · E ( L ) )= K X ′ . The third condition in Definition 2.10 is trivial since ǫ ∗ , σ and ǫ ∗ all preserve the effectiveclass. (cid:3) Proof of Lemma 4.3.
First, let C , . . . , C N be irreducible curves of the first kind thatare permuted by σ . It follows from Castelnuovo’s contraction theorem that there exist arational surface X ′ and a birational morphism ǫ : X → X ′ such that ǫ contracts C , . . . , C N and is an isomorphism outside C ∪ · · · ∪ C N . Let σ ′ = ǫ ∗ σǫ ∗ . By Lemma 4.4, σ ′ is aCremona isometry on Pic X ′ and thus we obtain an equation ( X ′ , σ ′ ). It is clear, byconstruction, that ǫ, X ′ , σ ′ satisfy the conditions in Definition 4.2.Next we show the converse. Let ǫ, X ′ , σ ′ satisfy the conditions in Definition 4.2:Pic X σ / / ǫ ∗ (cid:15) (cid:15) Pic X ǫ ∗ (cid:15) (cid:15) Pic X ′ σ ′ / / Pic X ′ and take an exceptional curve of the first kind C that is contracted by ǫ . Since ǫ ∗ σ ℓ [ C ] = σ ′ ℓ ǫ ∗ [ C ] = 0 ,σ ℓ C is contracted by ǫ for all ℓ . However, ǫ contracts only a finite number of curves.Thus, there exists N > σ N C = C . Hence σ acts as a permutation on ւ ❅❅❅❅❅ { y = ∞} { x = ∞} C ❅❅❅❅❅❅❅❅❅❅ t t ց Figure 4.
The mapping in Example 4.5 permutes two axes x and y . Ifwe consider this mapping on the upper surface, it has two minimizations. { C, σC, . . . , σ N − C } . Since these curves are exceptional curves of the first kind and arecontracted by σ , they are mutually disjoint. (cid:3) As in the autonomous case, one must first verify if there are such contractible curves.If so, then we obtain an equation ( X ′ , σ ′ ) by contracting these curves. It is clear that thedegree growth of ( X, σ ) is the same as that of ( X ′ , σ ′ ). Replacing ( X, σ ) with ( X ′ , σ ′ ) andrepeating this procedure, we finally obtain a surface on which the equation is minimal.As shown in the following example, a minimization is not unique in general. Example 4.5.
Let X be the surface obtained by blowing up P × P at ( ∞ , ∞ ), and let ϕ ( x, y ) = ( y, x ). It is clear that ϕ is an automorphism on X . X has three exceptional curves of the first kind: C, { x = ∞} and { y = ∞} (Figure 4).This mapping has two minimizations.The first possibility is P × P . Since C is fixed by ϕ , we can minimize ϕ from X to P × P , and it is trivial that ϕ is an automorphism on P × P .The second possibility is P . Since two curves { y = ∞} and { x = ∞} are permuted by ϕ , we can minimize ϕ from X to P by contracting these curves.We will show in Proposition 4.12 (for the integrable case) and Proposition 4.18 (in thenonintegrable case) that if the degree growth is unbounded, i.e. σ is of infinite order, thenthe minimization is unique.4.1. Integrable case.
In this subsection, we consider a minimization in the case of in-tegrable equations.In the autonomous case, a minimal space of initial conditions is always an elliptic surfaceand the equation preserves the elliptic fibration [13]. Thus, the theory of rational ellipticsurfaces is relevant to the problem of minimization. In the nonautonomous case, however, space of initial conditions does not have an elliptic fibration in general [35]. Therefore,we will have to find the contractible curves in Lemma 4.3 in a different way.The following is our main theorem in this paper: Theorem 4.6.
Consider an equation ( X, σ ) and assume that its degree growth is qua-dratic. Then, this equation can be minimized to a generalized Halphen surface.In particular, if a mapping of the plane with unbounded degree growth and zero algebraicentropy has a space of initial conditions, then it must be one of the discrete Painlev´eequations. Note that in this paper, “discrete Painlev´e equation” should be understood in Sakai’ssense, as defined in Definition 2.18.
Lemma 4.7.
Let X be a rational surface with ρ ( X ) = 10 . Then X is a generalizedHalphen surface if and only if − K X is nef.Proof. Suppose X is a generalized Halphen surface. Let C ⊂ X be an irreducible curve.If C is a component of − K X , then − K X · C = 0, by definition. On the other hand, if C is not a component of − K X , then − K X · C ≥ − K X is effective. In both cases wehave − K X · C ≥ − K X is nef.Let us prove the converse. Suppose − K X is nef. Since ρ ( X ) = 10, we have ( − K X ) =0 and − K X is effective ([35], Proposition 2). Thus it is sufficient to show that everycomponent of − K X is orthogonal to − K X . Let P i a i D i ∈ | − K X | . Since − K X is nef, wehave a i D i · ( − K X ) ≥
0. Summing, we obtain X i a i D i · ( − K X ) ≥ . Since the left hand side is equal to ( − K X ) , D i · ( − K X ) must be 0 for all i . Hence X isa generalized Halphen surface. (cid:3) Lemma 4.8.
Let X be a basic rational surface and let σ be a Cremona isometry on Pic X with quadratic growth. Let v , v , v ∈ Pic Q X \ { } satisfy σv = v ,σv = v + v ,σv = v + v . Then, we have • v is isotropic, • either v or − v is nef, • v · K X = 0 .Proof. That v is isotropic follows immediately from Proposition 3.2 .Let e = ( e (0) , . . . , e ( r ) ) be a geometric basis. Then, by Proposition 3.2 and Lemma 3.7,there exists a ∈ Q × such that v = a lim n → + ∞ n σ n e (0) . Since e (0) is nef and σ preserves the nef cone (Lemma 2.12), n σ n e (0) is nef for all n .Therefore, Proposition A.16 implies that a v is nef. ince v · K X = ( σv ) · ( σK X )= ( v + v ) · K X = v · K X + v · K X , we have v · K X = 0. (cid:3) Note that while v , v above are not unique, v is unique up to scaling. v is determinedby Q v = Ker( σ Q − id) ∩ Im( σ Q − id) , where σ Q is the Q -extension of σ to Pic Q X . Definition 4.9.
Let us normalize v so that • v is nef, • v ∈ Pic X , • v is primitive in Pic X , i.e. if a rational number a satisfies av ∈ Pic X , then a isan integer.We shall call this v the normalized dominant eigenvector of σ . Lemma 4.10.
Let X be a rational surface of Picard number . If X has a Cremonaisometry which grows quadratically, then X must be a generalized Halphen surface and − K X coincides with the normalized dominant eigenvector.Proof. Let σ be a Cremona isometry on Pic X that grows quadratically and let v be thenormalized dominant eigenvector of σ . By Lemma 4.8, v is isotropic and v · K X = 0.However, K X is also isotropic since ρ ( X ) = 10. Therefore, by Lemma 3.6, v and K X arelinearly dependent. Since v and K X are both primitive in Pic X , we have v = ± K X .While v is nef by Lemma 4.8, K X cannot be nef since X is rational. Thus we have v = − K X and Lemma 4.7 implies that X is a generalized Halphen surface. (cid:3) The following lemma is the key to the proof of Theorem 4.6
Lemma 4.11.
Let X be a rational surface with ρ ( X ) > and let σ be a Cremonaisometry on Pic X with quadratic growth. Then, X is not minimal for the equation ( X, σ ) .Proof. We will try to find mutually disjoint exceptional curves of the first kind that arepermuted by σ (Lemma 4.3).Step 1Let v ∈ Pic X be the normalized dominant eigenvector of σ . We first show that v + K X is effective and nonzero.By the Riemann-Roch inequality, we have h ( v + K X ) + h ( v + K X ) ≥ v + K X ) · v = 1 . Using Serre duality we have h ( v + K X ) = h ( − v ) = 0 . Hence, h ( v + K X ) ≥ v + K X is effective. It immediately follows from ( v + K X ) =10 − ρ ( X ) < v + K X = 0.Step 2Let C = { C ⊂ X : irreducible | C · ( v + K X ) < } . e show that C is a nonempty finite set.By Step 1, we can express v + K X as v + K X = " ℓ X i =1 a i C i , where the C i are irreducible and a i >
0. Since ( v + K X ) <
0, at least one of C , . . . , C ℓ satisfy C i · ( v + K X ) <
0. Thus C is not empty.On the other hand, if an irreducible curve C is different from C , . . . , C ℓ , then it satisfies C · ( v + K X ) ≥
0. Hence C is finite.Step 3We show that if C ∈ C , then C = − , C · K X = − , C · v = 0 , C ∼ = P . By the genus formula, we have g a ( C ) = 1 + 12 C · ( C + K X )= 1 + 12 C + 12 C · ( v + K X ) − C · v . Since g a ( C ) ≥ C < C · ( v + K X ) < C · v ≥
0, the only possible case is g a ( C ) = 0 , C = − , C · ( v + K X ) = − , C · v = 0 . It follows from Proposition A.13 that C ∼ = P .Step 4Since σ is a Cremona isometry, Lemma 2.13 implies that σ acts on C as a permutation.Step 5Let C, C ′ ∈ C satisfy C = C . We show that C ∩ C ′ = ∅ .Let m = C · C ′ . Since ( C + C ′ ) · v = 0, Lemma 3.6 implies that0 ≥ ( C + C ′ ) = 2 m − m = 0 or m = 1. Assume that m = 1. In this case, v and C + C ′ areorthogonal and both isotropic. Thus, again by Lemma 3.6, there exists a ∈ Q × suchthat [ C + C ′ ] = av . Since v and [ C + C ′ ] are both primitive and effective, we have a = 1. On the other hand, since C and C ′ are two different components of v + K X , thereexists an effective class F such that [ C ] + [ C ′ ] + F = v + K X . Thus we have F = K X ,which is a contradiction since K X cannot be effective when X is rational. Hence we have C · C ′ = 0. (cid:3) Proof of Theorem 4.6.
Let X be a rational surface and let σ be a Cremona isometry onPic X with quadratic growth. We show that one can minimize σ from X to a generalizedHalphen surface.It follows from Proposition 3.8 that ρ ( X ) ≥
10. If ρ ( X ) = 10, then Lemma 4.10 impliesthat X is a generalized Halphen surface, and thus ( X, σ ) is a discrete Painlev´e equation.Consider the case ρ ( X ) >
10. By Lemma 4.11, the equation (
X, σ ) is not minimal. Let ǫ : X → X ′ be a minimization and let σ ′ = ǫ ∗ σǫ ∗ . The minimality of ( X ′ , σ ′ ) implies that ρ ( X ′ ) ≤
10. However, it follows from Proposition 3.8 that ρ ( X ′ ) ≥
10 since the degreegrows quadratically. Thus Lemma 4.10 implies that X is a generalized Halphen surfaceand hence the equation ( X ′ , σ ′ ) is a discrete Painlev´e equation. (cid:3) lthough the proofs of Lemma 4.11 and Theorem 4.6 define a program to minimize( X, σ ), it could be a little difficult to describe the set C in Step 2 of the proof of Lemma 4.11explicitly. The following proposition tells us how to find a minimization only by linearalgebra and, at the same time, shows the uniqueness of the minimization. Proposition 4.12.
Let X be a rational surface with ρ ( X ) = r + 1 > and σ a Cremonaisometry on Pic X that grows quadratically. Let ǫ : X → X ′ be a minimization of ( X, σ ) .Decompose ǫ into a composition of blow-ups ǫ = ǫ (1) ◦ · · · ◦ ǫ ( r − and let E ( i ) ∈ Pic X be the total transform of the exceptional class of ǫ ( i ) for i = 1 , . . . , r − .Let v ∈ Pic X be the normalized dominant eigenvector and e = ( e (0) , . . . , e ( r ) ) an arbitrarygeometric basis on Pic X . Then the set { E (1) , . . . E ( r − } can be written as (4.1) E = { E ∈ Pic X | E = − , E · v = 0 , E · K X = − , E · e (0) ≥ , ( v − E ) · e (0) ≥ } . In particular, a minimization ǫ : X → X ′ is unique.Proof. Step 1We show that E ( i ) ∈ E for i = 1 , . . . , r −
9. It is sufficient to show that ( v − E ( i ) ) · e (0) ≥ e (0) is nef and v + K X − E ( i ) = E (1) + · · · E ( i − + E ( i +1) + · · · E ( r − is effective, we have 0 ≤ ( v + K X − E ( i ) ) · e (0) = − v − E ( i ) ) · e (0) . Step 2Let E ∈ E . We show that E and K X + v − E are both effective.By the Riemann-Roch inequality, we have h ( E ) + h ( E ) ≥ E · ( E − K X ) = 1 . Using Serre duality, we have h ( E ) = h ( K X − E ) . Since ( K X − E ) · e (0) ≤ − ,K X − E is not effective and thus h ( K X − E ) = 0. Therefore we have h ( E ) > h ( K X + v − E ) ≥ K X + v − E ) · ( v − E ) − h ( K X + v − E )= 1 − h ( − v + E ) . It follows from ( − v + E ) · e (0) < h ( − v + E ) = 0. Thus we have h ( K X + v − E ) > E, E ′ ∈ E and E = E ′ , then E · E ′ = 0. It is important to notethat E, E ′ ∈ v ⊥ and that the intersection is semi-negative definite on v ⊥ and its kernel isgenerated by v .Let m = E · E ′ . Since 0 ≥ ( E ± E ′ ) = − ± m, we have m = 0 , ±
1. We can exclude the cases m = ± ssume that m = 1. In this case, E + E ′ is isotropic and thus there exists α such that E + E ′ = αv . However, this leads to the contradiction:0 = αv · K X = ( E + E ′ ) · K X = − . Assume that m = −
1. As in the case of m = 1, there exists α such that E − E ′ = αv .Since v is primitive, α is a nonzero integer. We may assume α >
0. Thus we have E ′ + ( K X + v − E ) = (1 − α ) v + K X . However, while the left hand side is effective, the right hand side is not. Hence we concludethat E · E ′ = 0.Step 4We show that E ⊂ { E (1) , . . . , E (9 − r ) } .Assume that there exists E ∈ E \ { E (1) , . . . , E (9 − r ) } . It follows from Steps 1 and 3 that E · E ( i ) = 0 for i = 1 , . . . , E ( r − . However, this leads to the contradiction: − E · K X = E · ( − v − E (1) − · · · − E ( r − ) = 0 . Step 5The uniqueness of the minimization follows from the fact that the set E does not dependon ǫ . (cid:3) The normalized dominant eigenvector v is determined by Z v = Ker( σ Q − id) ∩ Im( σ Q − id) ∩ Pic X and v · e (0) > . Thus, in principle, we can calculate v and therefore E explicitly. Hence, this propositionallows us to obtain ( X ′ , σ ′ ) from ( X, σ ) by mere linear algebra.
Example 4.13.
Let us consider the equation in Example 2.1.We have already constructed a space of initial conditions X n for this equation. Whilethe equation is integrable, X n has Picard number 11. Therefore, this space of initialconditions is not minimal and should have one contractible curve. Let us explicitly findthe contractible curve. Instead of introducing ι n and Φ as in Remark 2.7, we identify allPic X n by using the basis D (1) n , D (2) n , C (1) n , . . . , C (6) n , e C (1) n , e C (2) n , e C (3) n for all n .First we calculate the class v + K X n , where v ∈ Pic X n is the normalized dominanteigenvector. Using the matrix (1.8), we find that v is written as v = 3 D (1) + 3 D (2) + 2 C (1) + · · · + 2 C (6) − e C (1) − e C (2) − e C (3) . Let H x , H y ∈ Pic X n be the total transforms of the curves { x = const } and { y = const } ,respectively. Then, these classes can be written as H x = D (1) + C (2) + C (4) + C (6) , H y = D (2) + C (1) + C (3) + C (5) , and thus we have v = 3 H x + 3 H y − C (1) − · · · − C (6) − e C (1) − e C (2) − e C (3) . Since K X n = − D (1) − D (2) − C (1) − · · · − C (6) + e C (1) + e C (2) + e C (3) , we have v + K X = H x + H y − e C (1) − e C (2) − e C (3) . Next, let us find the contractible curve C ⊂ X n . Since C is the only contractible curvein X n , the class v + K X n must represent C . Thus, the above expression implies that the mage of C in P × P has bidegree (1 ,
1) and passes through Q (1) , Q (3) , Q (3) . A directcalculation shows that the image of C in P × P is { y n − x n = 2 a n − α } . In fact, one can find that ϕ n ( { y n − x n = 2 a n − α } ) = { y n +1 − x n +1 = 2 a n +1 − α } , which means that, in a sense, the curve { y − x = 2 a − α } is invariant under the equation.Contracting this curve, we obtain the minimal space of initial conditions.4.2. Nonintegrable case.
In this subsection, we consider a minimization of a space ofinitial conditions in the nonintegrable case. We will give in Proposition 4.17 a minimal-ity criterion for a space of initial conditions and we show in Proposition 4.18 that theminimization of a space of initial conditions is unique.In this subsection, we consider the following situation: • X : a basic rational surface with ρ ( X ) = r + 1 > • σ : a Cremona isometry on Pic X with exponential growth. • λ >
1: the maximum eigenvalue of σ . • v ∈ Pic R X : the dominant eigenvector of σ , which is isotropic. Lemma 4.14. v or − v is nef.Proof. The proof is the same as that of Lemma 4.8. Take a geometric basis e = ( e (0) , . . . , e ( r ) )and consider the limit lim n → + ∞ λ n σ n e (0) . (cid:3) As the sign can be changed at will, we may assume that v is nef.We shall use the following lemma throughout this subsection. Lemma 4.15.
The intersection number is negative definite on the lattice v ⊥ ∩ Pic X .Proof. Lemma 3.6 says that the intersection number has signature (0 , r −
1) on v ⊥ and itskernel is generated by v . However, a scalar multiple of v does not belong to Pic X since λ is irrational. Thus the intersection number is negative definite on v ⊥ ∩ Pic X . (cid:3) Lemma 4.16.
Two different exceptional curves of the first kind that belong to v ⊥ arealways orthogonal to each other.Proof. Let
C, C ′ ∈ v ⊥ be two different exceptional curves of the first kind. UsingLemma 4.15, we have 0 > ( C + C ′ ) = − C · C ′ and thus C · C ′ = 0. (cid:3) Proposition 4.17. ( X, σ ) is minimal if and only if there exist no exceptional curves ofthe first kind that are orthogonal to v .Proof. Suppose that (
X, σ ) is not minimal. Then there exist mutually disjoint exceptionalcurves of the first kind C , . . . , C N such that σ acts as a permutation on { C , . . . , C N } . Itis therefore sufficient to show that C · v = 0.Taking ℓ ∈ Z such that σ ℓ C = C , we have C · v = ( σ ℓ C ) · ( σ ℓ v ) = λ ℓ C · v, which shows that C · v = 0. e now show the converse. Let C be the set of the exceptional curves of the first kindthat are orthogonal to v , and assume that C is nonempty. It is clear that σ acts on C asa permutation. Since all elements in C are mutually disjoint by Lemma 4.16, it followsfrom Lemma 4.3 that ( X, σ ) is not minimal. (cid:3)
While the Picard numbers of the minimal spaces of initial conditions for integrable sys-tems are always 10, those of nonintegrable systems depend on the detail of the equations.Therefore, it is impossible to check the minimality only by the Picard number. We canonly say that the Picard numbers are greater than 10. However, Proposition 4.17 givesus a precise minimality criterion.The following proposition is an analogue of Proposition 4.12.
Proposition 4.18.
Let ǫ : X → X ′ be a minimization of ( X, σ ) . Decompose ǫ into acomposition of blow-ups ǫ = ǫ (1) ◦ · · · ◦ ǫ ( L ) and let E ( i ) ∈ Pic X be the total transform of the exceptional class of ǫ ( i ) for i =1 , . . . , L . Let e = ( e (0) , . . . , e ( r ) ) be an arbitrary geometric basis on Pic X . Then theset { E (1) , . . . E ( L ) } can be written as E = { E ∈ Pic X | E = − , E · v = 0 , E · K X = − , E · e (0) ≥ } . In particular, the minimization ǫ : X → X ′ is unique.Proof. It is clear that E i ∈ E for i = 1 , . . . , L . We show that E ⊂ { E , . . . , E L } .Step 1We show that every element E ∈ E is effective. Using the Riemann-Roch inequalityand Serre duality, we have h ( E ) ≥ − h ( E ) = 1 − h ( K X − E ) . It follows from e (0) · ( K X − E ) < h ( K X − E ) = 0. Thus E is effective.Step 2We show that two different elements E, E ′ ∈ E are orthogonal to each other. Since theintersection number is negative definite on v ⊥ ∩ Pic X , we have0 > ( E ± E ′ ) = − ± E · E ′ , and thus E · E ′ = 0.Step 3Assume that there exists E ∈ E \ { E , . . . , E L } . Let E ′ = ǫ ∗ E and v ′ = ǫ ∗ v . Since E · E i = 0 by Step 2, we have E ′ = − , E ′ · K X ′ = − , E ′ · v ′ = 0 . Since E is effective, so is E ′ . Let us express E ′ as a sum of irreducible curves: E ′ = ℓ X j =1 a i [ C i ] . We show that there exists at least one j such that C j is an exceptional curve of thefirst kind. Since E ′ · v ′ = 0 and v ′ is nef, we have C j · v ′ = 0 for j = 1 , . . . , ℓ . Since theintersection number is negative definite on v ′⊥ ∩ Pic X ′ , C j are all negative. By the genusformula, we have g a ( C j ) = 1 + 12 C j + 12 C j · K X ′ . ultiplying with a j and summing, we obtain X j a j g a ( C j ) = X j a j + 12 X j a j C j + 12 X j a j C j · K X ′ . Using P j a j C j = E ′ and E ′ · K X ′ = −
1, we have X j a j (2 + C j − g a ( C j )) = 1 . If C j − g a ( C j ) ≤ − j , then the left hand side is not positive, which is a contra-diction. Therefore, there is at least one j such that C j − g a ( C j ) ≥ −
1. Since C j < C j = − , g a ( C j ) = 0 . Thus C j is an exceptional curve of the first kind.Since C j · v ′ = 0, Proposition 4.17 implies that ( X ′ , σ ′ ) is not minimal, which is acontradiction. Hence, we have E = { E , . . . , E L } . (cid:3) Conclusion
In this paper, we studied nonautonomous mappings of the plane with spaces of initialconditions and unbounded degree growth by means of spaces of initial conditions. Es-pecially, Theorem 4.6 shows that if an integrable mapping of the plane with unboundeddegree growth has a space of initial conditions, then it must be one of the discrete Painlev´eequations. Since all discrete Painlev´e equations have already been classified by Sakai [35],this means we have finished the classification of integrable mappings of the plane witha space of initial conditions and unbounded degree growth. Moreover, we have given aconcrete procedure to minimize a space of initial conditions to a generalized Halphen sur-face in the integrable case, as well as a general procedure to minimize the space of initialconditions for nonintegrable mappings.
Acknowledgement
I wish to express my gratitude to Professors R. Willox, A. Ramani and B. Grammaticosfor many useful discussions and comments. I would also like to thank A. Ramani forproviding me with the nice examples in Examples 1.5 and 2.1. This work was partiallysupported by a Grant-in-Aid for Scientific Research of Japan Society for the Promotion ofScience (25 · Appendix A. Algebraic surfaces
In this appendix we specify the notation used throughout the paper and we recall somebasic results on algebraic surfaces. We shall not give the proofs of propositions since thesecan be found in many textbooks such as [18, 2, 1].In this paper, a surface means a smooth projective variety of dimension 2 over C , whichbecomes a compact complex manifold of dimension 2. Notation A.1. • ∼ : the linear equivalence of divisors, which is the same as the numerical equiva-lence if the surface is rational. • [ D ]: the linear equivalence class of D . • Pic X : the Picard group of X . Pic Q X = Pic X ⊗ Q , Pic R X = Pic X ⊗ R , Pic C X = Pic X ⊗ C . • ρ ( X ): the Picard number of X , which is equal to dim Q (Pic Q X ) if X is rational. • H i ( X, D ) = H i ( D ): the i -th cohomology group of the divisor D (or its class). • h i ( D ) = h i ( X, D ) = dim C H i ( X, D ). • | F | : the linear system of F . • dim | F | = h ( F ) − F is effective. • K X : the canonical class of X . • D · D : the intersection number of the divisors D and D (or their classes). • X → Y : a morphism from X to Y . • X Y : a birational map from X to Y . • O P (1): the class of lines in P , which is a generator of Pic P as a Z -module. • PGL(3): the set of projective linear transformations on P , which coincides withthe group of automorphisms on P . • g a ( C ) = dim C H ( C, O C ): the arithmetic genus of an irreducible curve C . Theorem A.2 (Hodge index theorem) . The intersection number on a surface X hassignature (1 , ρ ( X ) − . Definition A.3.
Let f : X Y be a birational map and let π : e X → X be a resolutionof the indeterminacies of f : e X π (cid:15) (cid:15) g (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ X f / / Y. Then, f ∗ and f ∗ are defined by f ∗ = g ∗ π ∗ , f ∗ = π ∗ g ∗ . It is known that these linear maps do not depend on the choice of π .These linear maps do not preserve the intersection number in general. Proposition A.4.
The above f ∗ and f ∗ preserve the set of effective classes. Remark A.5. If f : Z → Y and g : Y → X are birational morphisms, then( g ◦ f ) ∗ = g ∗ f ∗ and ( g ◦ f ) ∗ = f ∗ g ∗ . However, these relations do not hold if f, g are simply birational maps.
Definition A.6 (degree) . Using the homogeneous coordinate ( z : z : z ) ∈ P , abirational automorphism ϕ on P can be written as ϕ ( z : z : z ) = ( ϕ ( z , z , z ) : ϕ ( z , z , z ) : ϕ ( z , z , z )) , where ϕ , ϕ , ϕ are homogeneous polynomials of z , z , z of the same degree with nocommon factors. The degree of ϕ (as a birational automorphism on P ) is defined bydeg ϕ i .It is known that deg ϕ can be calculated as follows:deg ϕ = ( ϕ ∗ O P (1)) · O P (1) = ( ϕ ∗ O P (1)) · O P (1) . Definition A.7 (basic rational surface) . A rational surface that admits a birationalmorphism to P is called a basic rational surface . This means that a basic rational surfacecan always be obtained by a finite number of blow-ups of P . efinition A.8 (geometric basis [10]) . Let X be a basic rational surface and ( e (0) , . . . , e ( r ) )be a Z -basis on Pic X . Then ( e (0) , . . . , e ( r ) ) is said to be a geometric basis if there is acomposition of blow-ups π = π (1) ◦ · · · ◦ π ( r ) : X → P such that e (0) = π ∗ O P (1) and e ( i ) is the total transform of the class of the exceptional curve of π ( i ) for i = 1 , . . . , r .Note that e ( i ) − e ( j ) ( i, j > , i = j ) is effective if and only if i < j and the center of the j -th blow-up is infinitely near that of the i -th blow-up.For any geometric basis ( e (0) , . . . , e ( r ) ) we have K X = − e (0) + e (1) + · · · + e ( r ) , e ( i ) · e ( j ) = i = j = 0) − i = j = 0)0 ( i = j ) . A birational morphism to P is determined only by its geometric basis up to automor-phism on P , i.e. if two birational morphisms π, π ′ : X → P give the same geometric basison Pic X , then there exists f ∈ PGL(3) such that π ′ = f ◦ π . In fact, these birationalmorphisms are determined only by e (0) , since the set { e (1) , . . . , e ( r ) } is determined by (cid:8) e (1) , . . . , e ( r ) (cid:9) = (cid:8) F ∈ Pic X | F = − , F · e (0) = 0 , F : effective (cid:9) . Proposition A.9.
Let C , . . . , C m be irreducible curves in a surface X such that thematrix ( C i · C j ) ij is negative definite. Then, for all nonnegative integers a , . . . , a m , wehave h X i a i C i ! = 1 . In particular, if an irreducible curve C has a negative self-intersection, then h ( mC ) = 1 for m ≥ and thus the class [ C ] cannot be written as a nontrivial sum of effective classes. Theorem A.10 (Riemann-Roch) . Let F ∈ Pic X . Then h ( F ) − h ( F ) + h ( F ) = χ ( O X ) + 12 F · ( F − K X ) , where we denote by χ ( O X ) the Euler-Poincar´e characteristic. Since h ( F ) is not negative,we have h ( F ) + h ( F ) ≥ χ ( O X ) + 12 F · ( F − K X ) , which is called the Riemann-Roch inequality . If X is rational, then χ ( O X ) = 1 . Thereforewe have h ( F ) − h ( F ) + h ( F ) = 1 + 12 F · ( F − K X ) and h ( F ) + h ( F ) ≥ F · ( F − K X ) . Theorem A.11 (Serre duality) . Let F ∈ Pic X . Then h i ( F ) = h − i ( K X − F ) for i = 0 , , . Theorem A.12 (genus formula) . Let C ⊂ X be an irreducible curve. Then g a ( C ) = 1 + 12 C · ( C + K X ) , where g a ( C ) = h ( C, O C ) is the arithmetic genus of C . roposition A.13. Let C be a (possibly singular) irreducible curve. Then g a ( C ) = 0 ifand only if C is isomorphic to P . Definition A.14 (exceptional curve of the first kind) . An irreducible curve C ⊂ X is called an exceptional curve of the first kind if C is isomorphic to P and C = − C = C · K X = − Theorem A.15 (Castelnuovo’s contraction theorem) . Let X be a surface and C ⊂ X an exceptional curve of the first kind. Then there exist a surface X ′ and a birationalmorphism π : X → X ′ such that π ( C ) is a point in X ′ and π is an isomorphism from X \ C to X ′ \ π ( C ) . This procedure is called a blow-down . In other words, we can contractan exceptional curve of the first kind by a blow-down. Definition A.16 (nef) . A divisor D on a surface X is nef if it satisfies C · D ≥ C ⊂ X . A class F ∈ Pic X (or F ∈ Pic R X ) is said to be nefif it satisfies C · F ≥ C ⊂ X .It is known that the self-intersection of a nef class is always nonnegative. It is alsoknown that K X cannot be nef for a rational surface X .The set { F ∈ Pic R X | F : nef } is a closed convex cone in Pic R X , i.e. • if F, F ′ are nef, then so is F + F ′ , • if F is nef and a >
0, then aF is also nef, • the above set is a closed set in Pic R X .The set of all nef classes in Pic R X is called the nef cone of X . Theorem A.17 (Nagata [29]) . If a rational surface has infinitely many exceptional curvesof the first kind, then it is a basic rational surface.
Appendix B. Proof of Lemma 3.7
Proof of Lemma 3.7.
Let us extend ( − , − ) to a Hermitian form on V C .We show (1) in Steps 1–9.Step 1First let us consider the case where f has an eigenvalue whose modulus is not 1.If λ is an eigenvalue with | λ | 6 = 1 and v its corresponding eigenvector, then v is alwaysisotropic since ( v, v ) = ( f v, f v )= | λ | ( v, v ) . Step 2Let us show that if λ is an eigenvalue whose modulus is not 1, then λ is simple.Let v be the corresponding eigenvector and assume λ is not simple. Then there exists w , linearly independent of v , such that f w = λw or f w = λw + v. n the first case, v and w are orthogonal to each other since( v, w ) = | λ | ( v, w ) , which, together with ( v, v ) = ( w, w ) = 0, contradicts Lemma 3.6. Let us consider thesecond case. Since ( v, w ) = ( λv, λw + v )= | λ | ( v, w ) , we have ( v, w ) = 0 . In the same way we find ( w, w ) = | λ | ( w, w )and thus ( v, v ) = ( v, w ) = ( w, w ) = 0 , which again contradicts Lemma 3.6.Step 3We show that if λ , λ are different eigenvalues of f with | λ i | 6 = 1, then λ = 1 /λ . Inparticular, λ i must be real numbers.Let v , v be the corresponding eigenvectors. These vectors are both isotropic by Step 1,and linearly independent since they are eigenvectors corresponding to different eigenvalues.Thus it follows from Lemma 3.6 that ( v , v ) = 0. Since( v , v ) = λ λ ( v , v ) , we have λ λ = 1 . Step 4We show that if f has an eigenvalue whose modulus is not 1, then f is diagonalizable.We already know that such eigenvalues are simple.We therefore consider an eigenvalue µ of modulus 1 and assume vectors u , u satisfy f u = µu ,f u = µu + u . Since ( u , u ) = ( u , u ) + µ ( u , u ) ,u is isotropic. In the same way we find( v, u ) = λµ ( v, u )and thus ( v, v ) = ( v, u ) = ( u , u ) = 0 , which contradicts Lemma 3.6. Hence, f is diagonalizable.Step 5We show that if λ ∈ R \ {± } is an eigenvalue of f , then so is 1 /λ .Assume that 1 /λ is not an eigenvalue. Then, since all eigenvalues except λ have modulus1 (Step 3) and f is diagonalizable (Step 4), there exists a basis ( v, u , . . . , u r ) of V C suchthat f v = λv,f u i = µ i u i , here | µ i | = 1. Since ( v, u i ) = λµ i ( v, u i ) , we have ( v, u i ) = 0. Therefore, using Step 1, we obtain that v is orthogonal to all elementsin V C . However, this contradicts the nondegeneratedness of ( − , − ).Steps 1–5 show that if f has an eigenvalue whose modulus is not 1, then the Jordannormal form of f is (3.3). From now on, let us consider the case where all eigenvalues of f have modulus 1.Step 6It is clear that if f is diagonalizable, then its Jordan normal form is (3.1). Thus it issufficient to show that if f is not diagonalizable, then its Jordan normal form is (3.2).Step 7We show that the size of each Jordan block is at most 3.Assume that linearly independent vectors v , v , v , v satisfy f v = νv ,f v = νv + v ,f v = νv + v ,f v = νv + v . Using ( v , v i ) = ( v , v i ) + ν ( v , v i − ) , we have ( v , v i − ) = 0 for i = 2 , ,
4. Since( v , v ) = ( νv + v , νv + v )= ( v , v ) + ν ( v , v ) ,v is isotropic, which contradicts Lemma 3.6.Step 8We show that f has only one Jordan block whose size is greater than 1. In particular,the corresponding eigenvalue is ± µ, ν be eigenvalues of modulus 1 and let pairwise-linearly independent vectors v , v and w , w satisfy f v = µv , f w = νw ,f v = µv + v , f w = νw + w . It is sufficient to show that v and w are linearly dependent.The same calculation as in Step 4 implies that v and w are both isotropic. Therefore,since ( v , w ) = µν ( v , w ) , it follows from Lemma 3.6 that µ = ν . However, using( v , w ) = ( v , w ) + µ ( v , w ) , we have ( v , w ) = 0. Hence Lemma 3.6 shows that v and w are linearly dependent.Step 9Finally we show that the size of the Jordan block in Step 8 is exactly 3. It is sufficientto show that the size is not 2. ssume that ( v , v , u , . . . , u r − ) is a basis on V C such that f v = νv ,f v = νv + v ,f u i = µ i u i . Moreover, we can take v and v in V since ν = ±
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