Alexander Stokes
email: stokes@ms.u-tokyo.ac.jp.

Takafumi Mase

Ralph Willox

Basile Grammaticos

FULL DEAUTONOMISATION BY SINGULARITY CONFINEMENT AS AN INTEGRABILITY TEST

Introduction

The motivation for this paper stems from a number of observations of quite a remarkable phenomenon which arises in the context of the singularity analysis of birational mappings of the plane and which is related to their degree growth and, ultimately, to their integrability. This phenomenon manifests itself as the dynamical degree of a mapping being reflected in the characteristic polynomial for certain recurrences that the coefficients in the mapping have to obey for the singularity structure of the mapping to remain unchanged under a sufficiently general deautonomisation.

The first observations of this phenomenon can be traced back to so-called late-confining versions of discrete Painlevé equations [START_REF] Hietarina | Discrete Painlevé I and singularity confinement in projective space[END_REF], where singularities are still confined but not at the first opportunity. For non-autonomous mappings obtained through this kind of late confinement, as opposed to genuine discrete Painlevé equations, the dependence of the coefficients in the mapping on the independent variable ceases to be additive, multiplicative or through the arguments of elliptic functions, but rather the coefficients satisfy a recurrence relation whose characteristic polynomial has the dynamical degree of the mapping as a root. Outside the context of late confinement, it has been observed in many examples [START_REF] Mase | Deautonomization by singularity confinement: an algebro-geometric justification[END_REF][START_REF] Ramani | The redemption of singularity confinement[END_REF][START_REF] Grammaticos | Singularity confinement and fulldeautonomisation: a discrete integrability criterion[END_REF] that the minimal polynomial governing the degree growth of the equation also appears in confinement conditions on coefficients of the deautonomised mapping, though sometimes it is necessary to add to the mapping extra terms which do not affect the singularity patterns.

This relationship between dynamical degree and confinement conditions was first conjectured in [START_REF] Mase | Deautonomization by singularity confinement: an algebro-geometric justification[END_REF] and, based on these and subsequent observations, the method of full deautonomisation by singularity confinement was introduced in [START_REF] Grammaticos | Singularity confinement and fulldeautonomisation: a discrete integrability criterion[END_REF][START_REF] Ramani | The redemption of singularity confinement[END_REF] as a remarkably effective integrability detector, which works even when other methods based on singularity analysis [START_REF] Halburd | Elementary calculations of degree growth and entropy for discrete equations[END_REF][START_REF] Mase | Singularity confinement as an integrability criterion[END_REF][START_REF] Ramani | Calculating algebraic entropies: an express method[END_REF] fail. This correspondence between coefficient evolution and degree growth clearly requires explanation, and it is this problem that we address in this paper.

Singularity confinement was proposed in [START_REF] Grammaticos | Do integrable mappings have the Painlevé property?[END_REF] as a discrete counterpart to the Painlevé property of ordinary differential equations. The Painlevé property requires that if a solution develops a movable singularity, i.e. one whose location in the complex plane depends on initial conditions, then this does not induce multivaluedness which would obstruct the meaningful definition of a general solution for the equation. In the discrete case we are interested similarly in singularities which appear spontaneously when iterating the system, and require them to disappear after finitely many further iterations, i.e. the singular behaviour is confined to finitely many iterates and does not propagate. In the years after its introduction singularity confinement has been used to great effect as an integrability criterion, in particular to isolate deautonomised versions of known autonomous integrable systems. In particular, many discrete Painlevé equations were found [START_REF] Ramani | Discrete versions of the Painlevé equations[END_REF][START_REF] Grammaticos | Discrete Painlevé equations: derivation and properties[END_REF] by applying the method to Quispel-Roberts-Thompson (QRT) mappings [START_REF] Quispel | Integrable mappings and soliton equations[END_REF][START_REF] Quispel | Integrable mappings and soliton equations II[END_REF].

It is important to note, however, that singularity confinement is not a necessary criterion for integrability, in that there exist mappings with unconfined singularities which are integrable through linearisation. This is true even in the continuous case where there are linearisable differential equations which do not have the Painlevé property [START_REF] Ramani | Integrable systems without the Painlevé property[END_REF]. But most importantly, it is also not a sufficient condition since there exist examples in which all singularities are confined but the dynamics are chaotic, the most famous example being that due to Hietarinta and Viallet [START_REF] Hietarina | Singularity confinement and chaos in discrete systems[END_REF]. We shall work with a definition of integrability of birational mappings in terms of their dynamical degree, or equivalently algebraic entropy [START_REF] Bellon | Algebraic entropy[END_REF], following the idea that integrability in the discrete case is associated with slow growth in complexity under the dynamics, as evident in Arnol'd's notion of topological entropy [START_REF] Arnol | d, Dynamics of complexity of intersections[END_REF][START_REF] Arnol | Dynamics of intersections, Analysis, et cetera[END_REF], Veselov's ideas of polynomial growth of certain characteristics [START_REF] Veselov | Growth and integrability in the dynamics of mappings[END_REF], and related notions of entropy considered by Friedland [START_REF] Friedland | Entropy of polynomial and rational maps[END_REF] following from work with Milnor [START_REF] Friedland | Dynamical properties of plane polynomial automorphisms[END_REF]. Recall that for a second-order equation defining a birational mapping φ : (x n , y n ) → (x n+1 , y n+1 ), the limit lim k→∞ deg φ k 1/k is called the (first) dynamical degree of the mapping, in which deg φ k is the degree as a rational function of initial conditions of the k-times composition of φ. The limit lim k→∞ 1 k log deg φ k is used mainly in the integrable systems community under the name 'algebraic entropy' [START_REF] Bellon | Algebraic entropy[END_REF]. These concepts can be similarly defined for non-autonomous equations defining families of birational mappings, and we say that a mapping, autonomous or not, is integrable if its algebraic entropy is zero or if its dynamical degree is one.

In the years since the realisation that singularity confinement is not sufficient for integrability, thankfully, a complete classification of mappings of the plane was obtained, according to their degree growth and whether they have the singularity confinement property in the sense of a space of initial conditions, both in the autonomous case by Diller and Favre [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] and in the nonautonomous case by one of the authors [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF]. In particular if a non-autonomous mapping with the singularity confinement property has unbounded degree growth, then it must either fit into the framework of discrete Painlevé equations [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF] or be non-integrable. Therefore for singularity confinement to function as an integrability test, it needs only be combined with something that reliably detects the dynamical degree. We will show that full deautonomisation does exactly this.

We will illustrate singularity confinement and the phenomena described above through the example

x n+1 + x n-1 = 1 x 2 n , (1.1)
which is a QRT mapping belonging to Class I in the classification of [START_REF] Ramani | On the autonomous limit discrete Painlevé equations[END_REF] (see also [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equations[END_REF] for a derivation and [START_REF] Grammaticos | Full-deautonomisation of a class of second-order mappings in ancillary form[END_REF]Appendix] for an easily consultable list). The definition of the singularity confinement property for such an equation, when recast as a mapping of the projective plane P 2 , is that it is birationally conjugate to an automorphism. I.e., that one can perform a sequence of blow-ups and possibly blow-downs to obtain a rational surface on which the mapping is regularised as an automorphism. Phrased on the level of the equation itself, as it originally was in [START_REF] Grammaticos | Do integrable mappings have the Painlevé property?[END_REF], a singularity is constituted by an iteration from some initial condition that results in a loss of a degree of freedom, in the sense that the iterate takes a value at which the inverse mapping is undefined. As already mentioned above, 'singularity confinement' refers to the situation where finitely many further iterations lead the singularity to disappear, with the lost degree of freedom being recovered. For example, while iterating the equation above, if some iterate x n takes the value 0 while x n-1 = u is some finite constant, then the value of the subsequent iterate is x n+1 = ∞ irrespective of the precise value of u. This constitutes a loss of degree of freedom in the sense that the one-parameter family of pairs (x n-1 , x n ) = (u, 0) all lead to the same value (x n , x n+1 ) = (0, ∞). In terms of the iteration mapping (x n-1 , x n ) → (x n , x n+1 ), this is nothing but the curve given by x n = 0 being contracted to a point, i.e. a curve being blown down by the mapping.

Computing further we find that x n+2 = 0 but then, crucially, see that the value of the next iterate x n+3 = 1 x 2 n+2 -x n+1 is not defined, with the point (x n+1 , x n+2 ) = (∞, 0) being an indeterminacy of the mapping. The way in which singularity confinement is verified, is to take x n = ε for some small parameter ε which leads to

x n-1 = u, x n = ε, x n+1 = 1 ε 2 -u, x n+2 = -ε + O(ε 2 ), x n+3 = u + O(ε). (1.2)
Then, defining the values of iterates as the limits of the above as ε → 0, we have

x n-1 = u, x n = 0, x n+1 = ∞ 2 , x n+2 = 0, x n+3 = u, (1.3)
in which x n+1 = ∞ 2 means the iterate is of order 1/ε 2 as above. In singularity confinement parlance we say that the lost degree of freedom is recovered through the reappearance of u in the value of x n+3 , and that the singularity is confined. The results of the above calculation are then summarized by saying that equation (1.1) admits the (confined) singularity pattern {0, ∞ 2 , 0}.

To demonstrate the use of singularity confinement in the deautonomisation procedure, let us consider the following generalisation of mapping (1.1)

x n+1 + x n-1 = a n + b n x n x 2 n , (1.4)
where the n-dependence of the functions a n , b n is to be determined by requiring that the singularity structure of (1.4) is the same as in the autonomous case. As above, some iterate x n taking the value 0 constitutes a singularity of the mapping (1.4), so we let x n = ε and x n-1 = u, and compute

x n = ε, x n+1 = a n ε 2 + b n ε -u, x n+2 = -ε + b n+1 a n ε 2 + O(ε 3 ), x n+3 = a n+2 -a n ε 2 -b n+2 - 2a n+2 b n+1 a n + b n 1 ε + O(1)
.

(1.5)

In order for the singularity to be confined in the same way as above, the singular part of the expansion of x n+3 in ε must vanish, so we must have a n+2 = a n for all n, as well as

b n+2 -2b n+1 + b n = 0. (1.6)
Solving these 'confinement' conditions we find

x n+1 + x n-1 = α + βn x n + (-1) n γ + δ x 2 n , (1.7)
for some constants α, β, γ, δ, which reduces to a known example of a discrete Painlevé equation [START_REF] Ramani | Discrete Painlevé equations: coalescences, limits and degeneracies[END_REF]. More relevant to the present paper is the fact that the characteristic polynomial of the linear recurrence (1.6) is t 2 -2t + 1, which has no roots larger than one.

In the deautonomisation which led to the mapping (1.7), we required that the evolution of coefficients be such that the deautonomised mapping admits the exact same confined singularity pattern {0, ∞ 2 , 0}. However it is possible to choose a n , b n such that the singularity associated with the value 0 is confined but after more iterations rather than at the earliest opportunity, a notion called late confinement [START_REF] Grammaticos | Singularity confinement and fulldeautonomisation: a discrete integrability criterion[END_REF]. In particular, calculating further along the lines above but assuming that the previous confinement conditions do not hold, we find

x n = ε, x n+1 = a n ε 2 + b n ε -u, x n+2 = -ε + b n+1 a n ε 2 + O(ε 3 ), x n+3 = a n+2 -a n ε 2 -b n+2 - 2a n+2 b n+1 a n + b n 1 ε + O(1), x n+4 = ε + a n b n+3 -a n+2 b n+1 + a n b n+1 a n (a n+2 -a n ) ε 2 + O(ε 3 ), x n+5 = A n ε 2 -B n + 2A n b n+1 a n + b n+3 a n -a n+2 1 ε + O(1), (1.8)
where

A n = a n+4 -a n+2 + a n and B n = b n+4 -2b n+3 + b n+2 -2b n+1 + b n .
Therefore we have an opportunity to confine the singularity at a later stage, by ensuring that the singular powers of ε in x n+5 vanish, such that we have singularity confinement with the singularity pattern {0, ∞ 2 , 0, ∞ 2 , 0}. In this case the confinement conditions are given by the pair of recurrences

a n+4 -a n+2 + a n = 0, b n+4 -2b n+3 + b n+2 -2b n+1 + b n = 0, (1.9)
whose characteristic polynomial

t 4 -t 2 + 1 t 4 -2t 3 + t 2 -2t + 1 , (1.10)
has six roots of modulus one, as well as a pair λ, λ -1 where

λ = 1 + √ 2 + -1 + 2 √ 2 2 ∼ = 1.8832. (1.11)
The mapping (1.4) with coefficients satisfying (1.9) but otherwise generic is indeed non-integrable with dynamical degree given by this largest root λ of the characteristic polynomial, which can be proved by a number of methods, e.g. [START_REF] Takenawa | Algebraic entropy and the space of initial values for discrete dynamical systems[END_REF][START_REF] Halburd | Elementary calculations of degree growth and entropy for discrete equations[END_REF][START_REF] Ramani | Calculating algebraic entropies: an express method[END_REF]. More generally, consider the mapping

x n+1 + x n-1 = 1 x 2m n , (1.12)
for some integer m ≥ 1, which is known [START_REF] Grammaticos | Singularity confinement and fulldeautonomisation: a discrete integrability criterion[END_REF][START_REF] Ramani | Calculating algebraic entropies: an express method[END_REF] to have dynamical degree m+ √ m 2 -1. Similar analysis as above shows that this mapping admits the confined singularity pattern 0, ∞ 2m , 0 . If we proceed to deautonomise, by taking

x n+1 + x n-1 = a n x 2m n , (1.13)
we see by similar calculations to those for the m = 1 case above, that for the pattern 0, ∞ 2m , 0 to persist we must require the confinement condition a n+2 = a n . Note that this is the same condition as for the m = 1 case which, most importantly, does not give any information on the dynamical degree of the mapping. However, in performing these calculations with x n = ε one notices that adding terms 1 x j , j = 1, . . . , 2m -1 to the right-hand side of (1.12) does not affect the leading behaviours x n+1 ∼ 1 ε 2m , x n+2 ∼ ε, and hence that there still is an opportunity to confine at the next iterate. Doing so leads to the 'full deautonomisation' of the mapping (1.12) given by

x n+1 + x n-1 = a (2m) n x 2m n + 2m-1 i=2 a (i) n x i n + b n x n . (1.14)
The confinement conditions on the coefficients a

(i)
n and b n required for this mapping to admit the singularity pattern 0, ∞ 2m , 0 are

a (i) n+2 = (-1) i a (i) n , i = 2, . . . , 2m, b n+2 -2mb n+1 + b n = 0. (1.15)
The characteristic polynomial for the recurrence satisfied by b n is

t 2 -2mt + 1, (1.16)
whose largest root λ = m + √ m 2 -1 coincides with the dynamical degree of the mapping. It is worth emphasizing that this is also the dynamical degree of the non-autonomous mapping with coefficients that obey (1.15).

The remarkable connection that is observed here between degree growth and evolution of coefficients is what we aim to explain in this paper. The key to this explanation is offered by an extension of the notion of 'period mapping', originally introduced for integrable mappings of the plane. In the non-autonomous case, the singularity confinement property of a family φ n of birational mappings of the plane is equivalent to the existence of a space of initial conditions: a family of rational surfaces to which the mappings lift to isomorphisms (cf. Definition 2.4). In the theory of rational surfaces associated with discrete Painlevé equations due to Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], there is a known bridge between parameter evolution and degree growth, namely the interaction between dynamics on the Picard lattice of the surfaces and a kind of period map, whose construction goes back to the work of Looijenga [START_REF] Looijenga | Rational surfaces with an anticanonical cycle[END_REF]. This bridge relies on, amongst other things, the fact that the surfaces providing the space of initial conditions for the mapping have an effective anticanonical divisor, i.e. the fact that they admit a zero-free rational two-form .

In fact the question of the existence of effective anticanonical and, more generally, antipluricanonical divisors (here meaning a divisor representing some negative multiple of the canonical class) appears frequently in the study of automorphisms of algebraic surfaces and their dynamics [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF][START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF][START_REF] Diller | Invariant curves for birational surface maps[END_REF][START_REF] Diller | Rational surface maps with invariant meromorphic two-forms[END_REF][START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Harbourne | Rational surfaces with infinite automorphism group and no antipluricanonical curve[END_REF][START_REF] Mcmullen | Dynamics on blow-ups of the projective plane[END_REF][START_REF] Zhang | Automorphism groups and anti-pluricanonical curves[END_REF]. In particular, a conjecture of Gizatullin (communicated to Harbourne by Dolgachev and Looijenga [START_REF] Harbourne | Rational surfaces with infinite automorphism group and no antipluricanonical curve[END_REF]) stated that if a rational surface X has an automorphism φ of infinite order, then X must have an effective antipluricanonical divisor. In all counterexamples to this conjecture presented by Harbourne [START_REF] Harbourne | Rational surfaces with infinite automorphism group and no antipluricanonical curve[END_REF], φ has zero entropy and the pair (X, φ) is non-minimal in the sense that there exists a birational morphism π : X → X ′ such that π • φ • π -1 is an automorphism of X ′ but π is not an isomorphism. McMullen then posed in [START_REF] Mcmullen | Dynamics on blow-ups of the projective plane[END_REF]Section 12] a refined version of the conjecture, by asking whether for minimal (X, φ), φ being of infinite order is sufficient to guarantee that X has an effective antipluricanonical divisor. For certain classes of φ coming from group actions the conjecture is true [START_REF] Zhang | Automorphism groups and anti-pluricanonical curves[END_REF], but these also have zero entropy, i.e. also correspond to integrable mappings. A negative answer to McMullen's question was given by Bedford and Kim [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] who constructed a family of mappings which give automorphisms of infinite order with nonzero entropy, on rational surfaces which admit no effective antipluricanonical divisor.

In this paper we show that in a large class of non-autonomous mappings, namely those that preserve rational two-forms (hence including all mappings of QRT-type), even in non-integrable cases the space of initial conditions can be chosen via minimisation so that the surfaces are anticanonical, i.e. have effective anticanonical divisors. Then a period map exists and can be used to show that the evolution of coefficients under deautonomisation must correspond to the dynamics on the Picard lattice and thus encodes the dynamical degree. The main results of the paper can be summarised in the following theorem, which provides a justification of the method of full deautonomisation by singularity confinement as an integrability test.

Theorem 1.1. If an autonomous mapping has a space of initial conditions and unbounded degree growth, and preserves a rational two-form, the period map parametrisation of the coefficients in a sufficient deautonomisation gives a linear system of difference equations for their evolution, whose characteristic polynomial has the dynamical degree of the mapping as a root.

There are some subtle technical points relating to the applicability of the method of full deautonomisation as an integrability detector that we should point out here. The first is that, when applying the method to an autonomous mapping which preserves a rational two-form, so whose space of initial conditions is within the scope of Theorem 2.23 on effectiveness of the anticanonical divisor, one must ascertain the extent to which the mapping must be generalised via the addition of extra terms in order for the confinement conditions to reliably detect the dynamical degree. While it is possible to define a sufficiently general deautonomisation in terms of families of surfaces forming the space of initial conditions (cf. Definition 2.30), in order to make it possible to address this question directly on the level of the equations themselves, we provide the reader with sufficiently general deautonomisations for QRT-type classes I-VI in Appendix B.

A second subtle point, which also has implications for the implementation of the method, arises on the level of geometry. The family of examples constructed by Bedford and Kim [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] show that a period map construction, as we will present here, might not be possible for just any, general, birational mapping with a space of initial conditions because the existence of an effective anticanonical divisor is not guaranteed in the general setting. Therefore, in such examples, any hope of reading off the dynamical degree for the mapping from confinement conditions on coefficients in the mapping must be given up. The problem being that finding a linearisation of the conditions on the coefficients that might lead to a characteristic polynomial that encodes the dynamical degree is pretty much impossible, as is to be expected given the absence of a period map parametrisation of the coefficients in the mapping. This is similar to cases where the anticanonical divisor is effective, but for which the geometry is such that the parametrisation of its coefficients by the period map is highly non-trivial. This is the case in, for example, elliptic discrete Painlevé equations or QRT-type mappings of Class VIII, though for these mappings there are ways to overcome these difficulties and obtain an appropriate parametrisation of coefficients and therefore a linearisation of the confinement conditions such that full deautonomisation by singularity confinement still works [START_REF] Grammaticos | Full-deautonomisation of a class of second-order mappings in ancillary form[END_REF].

1.1. Outline of the paper. Readers interested in the utilisation of the method as an integrability test can proceed first to Sections 3-6, and additionally see Appendix B for details of sufficiently general deautonomisations for QRT classes I-V, before referring to Section 2 for the geometric theory behind it. Readers interested in the geometry of the space of initial conditions may proceed to Section 2 for the setup and proofs of the main results relating to anticanonical divisors and the period map and to Appendix A for a local classification of the anticanonical divisors of rational surfaces associated with non-integrable mappings. They can then consult Sections 3-6 for concrete examples.

Geometry

In this Section we give a rigorous formulation and justification for the full deautonomisation procedure as a detector of the dynamical degree of a mapping that possesses a space of initial conditions. We first recall the definition of the singularity confinement property, for a secondorder mapping, in the sense of the existence of a space of initial conditions for the mapping, in both the autonomous and non-autonomous cases. We then review the known classifications of autonomous and non-autonomous mappings according to their degree growth and according to whether they have a space of initial conditions, and deduce which cases full deautonomisation should be shown to discern between. Next we outline the period map defined for complex rational surfaces with effective anticanonical divisor, which will be the main tool in our formulation. For this tool to be applicable in the cases we need, we must prove Theorem 2.23, which states that in a wide class of mappings -including deautonomised versions of all those in the classification of QRT-type mappings in [START_REF] Ramani | On the autonomous limit discrete Painlevé equations[END_REF] (or [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equations[END_REF]) -if a mapping with a space of initial conditions is non-integrable, then the surfaces forming this space still have effective anticanonical divisors. We then show that coefficients in a non-autonomous mapping can always be parametrised by appropriate computation of the period map, in a way that linearises their evolution and such that the characteristic polynomial of the linearised recurrences will have the dynamical degree of the mapping as a root.

Most of the setup for this Section is given in [24, Appendix A], but we recall the relevant parts here in order to make the present paper self-contained. In what follows we work over C and use the following notation and conventions:

• X : a smooth projective rational surface.

• Div(X) : the group of divisors on X.

• ∼ : linear equivalence of divisors.

• Pic(X) : the Picard group of X, which is isomorphic to Div(X)/ ∼. We often use the same symbol for an element of Pic(X) as for its corresponding linear equivalence class of divisors and we write the group operation on Pic(X) additively. • |F| : the linear system of F ∈ Pic(X).

• Pic + (X) = {F ∈ Pic(X) | F = [F ] where F ∈ Div(X) is effective}: the set of effective classes. • D 1 • D 2 :
the intersection number of the divisors D 1 and D 2 . This is well-defined on linear equivalence classes and we use the same notation for the intersection number

D 1 • D 2 of D 1 , D 2 ∈ Pic(X). • (D) 2 = (D) 2 : the self-intersection number of D ∈ Div(X) or D = [D] ∈ Pic(X).
• K X ∈ Pic(X) : the canonical bundle or canonical divisor class of X.

• div(ω) ∈ Div(X) : the divisor of a rational two-form ω on X, so [div(ω)] = K X .

• O P n (1) ∈ Pic(P n ) : the twisting sheaf corresponding to the class of a hyperplane in

P n . • 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 rank Pic(X) since X is rational.

• H i (X, D) : the i-th cohomology group of the divisor D on X.

• h i (D) = h i (X, D) = dim H i (X, D). • b 1 = rank H 1 (X; Z) : the first Betti number.
• h p,q = dim H q (X, Ω p ) : the Hodge numbers.

• g a (C) = dim H 1 (C, O C ) : the arithmetic genus of an irreducible curve C.

2.1. Spaces of initial conditions for birational mappings of the plane. The singularity confinement property of a second-order system of difference equations defining a birational mapping of the plane is equivalent to the existence of a space of initial conditions. The terminology 'space of initial conditions' or 'space of initial values' originates in Okamoto's work [START_REF] Okamoto | Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé[END_REF] using blow-ups to construct augmented phase spaces on which Hamiltonian forms of the Painlevé differential equations are regularised. Solutions of these equations can be globally defined by analytic continuation from any point in the space, hence the term space of initial conditions. The same terminology continues to be used in the discrete case to describe rational surfaces on which birational mappings are regularised in an analogous way. In the autonomous case this means that the birational mapping becomes an automorphism of the surface. In the non-autonomous case the n-dependence in the coefficients means that the equation becomes a family of mappings, for which a space of initial conditions is a family of rational surfaces between which the mappings become isomorphisms.

2.1.1. Autonomous case. Consider an autonomous second-order discrete equation

(x n+1 , y n+1 ) = (f (x n , y n ), g(x n , y n )), (2.1)
where f , g are rational functions of their arguments with coefficients independent of n, such that the mapping (x n , y n ) → (f (x n , y n ), g(x n , y n )) is birational. By taking (x n , y n ) as an affine chart we can extend this to a birational mapping

φ : P 2 P 2 , (2.2)
where here and from this point onwards we use dashed arrows to denote birational mappings between surfaces, with solid arrows being morphisms. Note that scalar three-point equations

x n+1 = F (x n , x n-1
) with F homographic in x n-1 , such as the examples considered in the previous section, can be brought to this form by setting y n = x n-1 .

Definition 2.1 (space of initial conditions for an autonomous mapping). We say that an autonomous equation defining a mapping φ as above has a space of initial conditions if there exists a rational surface X and a birational map π : X P 2 such that φ := π -1 • φ • π is an automorphism of X:

X X P 2 P 2 . φ π π φ
The automorphism φ then induces, by pullback and pushforward, linear transformations of Pic(X) which we denote by φ * and φ * = ( φ * ) -1 respectively. These maps are lattice automorphisms of Pic(X), i.e. Z-module automorphisms which preserve the symmetric bilinear form given by the intersection product, they fix K X and preserve effectiveness of divisor classes.

2.1.2. Non-autonomous case. In the non-autonomous case, consider a second-order discrete system (x n+1 , y n+1 ) = (f n (x n , y n ), g n (x n , y n )), defining a family of mappings

φ n : P 2 P 2 . (2.3)
In addition to a family of rational surfaces X n on which φ n conjugates to an isomorphism X n → X n+1 , for each value of n, in the non-autonomous case it is necessary to include in the definition of a space of initial conditions some extra conditions in order to formulate a general theory. This is because allowing for arbitrary n-dependence in the mappings and surfaces leads to many pathological examples and prevents us from making any meaningful statements in general; see [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF] for a detailed discussion of this. In order to state the definition we will work with for non-autonomous mappings we require the following.

Definition 2.2 (basic rational surface). A basic rational surface is a rational surface that admits P 2 as a minimal model, i.e. there exists a birational morphism from X to P 2 . Such a surface can be obtained from P 2 by a finite number of blow-ups, without the need for blow-downs.

Definition 2.3 (geometric basis). Let X be a basic rational surface and (e (0) , . . . , e (r) ) a Z-basis for Pic(X). We call (e (0) , . . . , e (r) ) a geometric basis if there exists a composition of blow-ups

π = π (1) • • • • • π (r) : X → P 2 such that e (0) = π * O P 2 (1)
, where e (i) is the class of the total transform (under

π (i+1) • • • • • π (r)
) of the exceptional curve of π (i) for i = 1, . . . , r. In this case we say that (e (0) , . . . , e (r) ) is the geometric basis of Pic(X) corresponding to π.

Definition 2.4 (space of initial conditions for a non-autonomous mapping; without blowdowns). A space of initial conditions for a non-autonomous mapping φ n : P 2 P 2 consists of sequences (X n ) n and (π n ) n , where each X n is a rational surface and π n is a birational morphism π n : X n → P 2 written as a sequence of blow-ups

π n = π (1) n • • • • • π (r)
n , such that the following conditions hold:

• The mappings φ n become isomorphisms φn := π -1 n+1 • φ n • π n as in Figure 1.

• Let e n = (e (0) n , . . . , e (r)
n ) be the geometric basis for Pic(X n ) corresponding to π n . Then the matrices of φ * n : Pic(X n+1 ) → Pic(X n ) with respect to these bases do not depend on n.

• The set of effective classes in Pic(X n ) in terms of the basis (e

(0) n , . . . , e (r) n ) does not depend on n, i.e. if i a (i) e (i) n ∈ Pic(X n ) is effective, then i a (i) e (i) k ∈ Pic(X k ) is effective for any k.
Using the geometric bases e n we may identify all Pic(X n ) with a single Z-module

Pic(X ) = r i=0 Ze (i) ,
(2.4)

• • • X n-1 X n X n+1 • • • • • • P 2 P 2 P 2 • • • , φn-1 π n-1 φn πn π n+1 φ n-1 φn Figure 1.
Space of initial conditions for a non-autonomous mapping; without blow-downs

via ι n : Pic(X ) -→ Pic(X n ), ι n (e (i) ) = e (i) n , (2.5)
where (for reasons that will become clear when we discuss deautonomisation) we use X to distinguish this module from the Picard group of a single surface X. Denote the map induced by φ * n , for any n, by Φ : Pic(X ) → Pic(X ), which is well-defined because of the conditions in Definition 2.4:

Pic(X ) Pic(X ) Pic(X n ) Pic(X n+1 ). ιn ι n+1 Φ φ * n
The canonical class in each Pic(X n ) is identified with the element

K X := ι -1 n (K Xn ) = -3e (0) + e (1) + • • • + e (r) , (2.6)
and the intersection form on Pic(X n ) can be used to equip Pic(X ) with the symmetric bilinear form defined by

e (i) • e (j) =      +1 (i = j = 0) -1 (i = j ̸ = 0) 0 (i ̸ = j).
(2.7)

We will refer to Pic(X ) equipped with this bilinear form as the Picard lattice of the space of initial conditions X n . Then Φ is a lattice automorphism of Pic(X ) that fixes the element K X . It also preserves effectiveness in the sense that if

F ∈ Pic(X ) is such that ι n+1 (F) is an effective class on X n+1 , then ι n • Φ(F) is an effective class on X n .
While all the examples we will study in Sections 3-6 will have spaces of initial conditions as defined above, our main general results will rely on the notion of minimality of a space of initial conditions, which requires us to give a more general definition allowing blow-downs in the construction of X n from P 2 . Definition 2.5 (space of initial conditions for a non-autonomous mapping; allowing blowdowns). A space of initial conditions for a non-autonomous mapping φ n , allowing blowdowns, consists of sequences (

X n ) n , (Y n ) n , (π n ) n , (ρ n ) n , and (σ n ) n , with
• X n a rational surface with birational morphism π n : X n → P 2 , written as a sequence of blow-ups as

π n = π (r) n • • • • • π (1)
n , • Y n a rational surface with birational morphism ρ n : Y n → P 2 , written as a sequence of blow-ups as

ρ n = ρ (s) n • • • • • ρ (1) n , • σ n : Y n → X n a birational morphism written as sequences of blow-ups as σ n = σ (s-r) n • • • • • σ (1)
n , such that the following conditions hold:

• The mappings φn := σ n+1 • ρ -1 n+1 • φ n • ρ n • σ -1 n : X n → X n+1 are isomorphisms as in Figure 2. • Let f n = (f (0) n , . . . , f (s)
n ) be the geometric basis for Pic(Y n ) corresponding to ρ n . Then the set of effective classes in Pic(Y n ) in terms of the basis f n does not depend on n.

• Let e n = (e (0) n , . . . , e (r) n ) be the geometric basis for Pic(X n ) corresponding to π n . Then the set of effective classes in Pic(X n ) in terms of the basis e n does not depend on n.

• Let F (j)
n ∈ Pic(Y n ) be the class of (the total transform of) the exceptional curve contracted by σ (j) n , for j = 1, . . . , s -r. Then the expression for F (j) n in terms of the basis f n does not depend on n.

• The expression for σ * n (e

(i)
n ) in the basis f n for Pic(Y n ) does not depend on n. • The matrices of φ * n : Pic(X n+1 ) → Pic(X n ) with respect to the bases e n+1 and e n do not depend on n.

• • • Y n-1 Y n Y n+1 • • • • • • X n-1 X n X n+1 • • • • • • P 2 P 2 P 2 • • • • • • P 2 P 2 P 2 • • • σ n-1 ρ n-1 σn ρn σ n+1 ρ n+1 φn-1 π n-1 πn φn π n+1 φ n-1 φn

Figure 2. Space of initial conditions for a non-autonomous mapping; allowing blow-downs

If a mapping has a space of initial conditions as above then we may still define the Picard lattice Pic(X ) by identification of all Pic(X n ) via the geometric bases corresponding to π n , with the lattice automorphism Φ of Pic(X ) defined as above.

Remark 2.6. The key feature of the more general Definition 2.5 of a space of initial conditions is that σ n may contract curves on Y n which ρ n does not, in which case the mapping π n+1 • φn •π -1 n : P 2 P 2 will not coincide with φ n , but rather be conjugate to it by a birational coordinate change on P 2 . 2.1.3. Minimisation of a space of initial conditions. Allowing for blow-downs in the definition of a space of initial conditions means we can introduce the notion of a minimal space of initial conditions for a given non-autonomous mapping. Definition 2.7 (minimisation of a space of initial conditions). Let φ n : P 2 P 2 be a nonautonomous mapping with a space of initial conditions allowing blow-downs as in Definition 2.5. A minimisation of the space of initial conditions consists of sequences (X

′ n ) n and (µ n ) n with • X ′ n a rational surface • a birational morphism µ n : X n → X ′ n which is a sequence of blow-ups µ n = µ (1) n •• • ••µ (r ′ ) n
such that the following conditions hold:

• For every n, the map φ′

n := σ -1 n+1 • φn • σ n : X ′ n → X ′ n+1 is an isomorphism. • Let e n = (e (0) n , . . . , e (r)
n ) be the geometric basis for Pic(X n ) corresponding to π n , and let E (k) n ∈ Pic(X n ) be the class of (the total transform of) the exceptional divisor contracted by µ

(k) n for k = 1, . . . , r ′ . Then the expression for E (k)
n in terms of the basis e n is independent of n.

• There exist birational morphisms π ′ n : X ′ n → P 2 such that the geometric bases corresponding to π ′ n satisfy the requirements for X ′ n to form a space of initial conditions for φ n in the sense of Definition 2.5.

We give a diagram showing the birational mappings involved in minimisation of a space of initial conditions in Figure 3.

• • • Y n-1 Y n Y n+1 • • • • • • X n-1 X n X n+1 • • • • • • X ′ n-1 X ′ n X ′ n+1 • • • • • • P 2 P 2 P 2 • • • • • • P 2 P 2 P 2 • • • • • • P 2 P 2 P 2 • • • ρ n-1 σ n-1 ρn σn ρ n+1 σ n+1 π n-1 µ n-1 πn µn π n+1 µ n+1 π ′ n-1 π ′ n π ′ n+1 φ n-1 φn Figure 3.
Minimisation of a space of initial conditions Definition 2.8 (minimal space of initial conditions). For a space of initial conditions X n , we say that a minimisation is trivial if the maps µ n are isomorphisms. If there does not exist any nontrivial minimisation, then we say the space of initial conditions is minimal.

Whether a space of initial conditions can be minimised or not is characterised by the following.

Lemma 2.9 ( [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF]). If there exist a finite set of elements of Pic(X ) corresponding to mutually disjoint exceptional curves of the first kind which are permuted by Φ, then blowing down these curves provides a nontrivial minimisation of the space of initial conditions.

2.2. Degree growth and entropy for birational mappings of the plane. Hereafter, the definition of integrability of a birational mapping of the plane we will work with is that its algebraic entropy vanishes, or equivalently that its dynamical degree is one. For a birational map f from P 2 to itself, written in homogeneous coordinates as

f : P 2 P 2 , (x 0 : x 1 : x 2 ) → (f 0 (x 0 , x 1 , x 2 ) : f 1 (x 0 , x 1 , x 2 ) : f 2 (x 0 , x 1 , x 2 )) , (2.8)
where f 0 , f 1 , f 2 are homogeneous polynomials of the same degree with no common factors, its degree deg f is defined as the common degree of the polynomials f i .

Definition 2.10 (dynamical degree, algebraic entropy [START_REF] Bellon | Algebraic entropy[END_REF]). For a mapping φ n : P 2 P 2 , the numbers

d = lim k→∞ deg φ (k) 1/k
, and ε = lim

k→∞ 1 k log deg φ (k) , (2.9)
where

φ (k) = φ n+k-1 • • • • φ n+1 • φ n ,
are called the dynamical degree and algebraic entropy [START_REF] Bellon | Algebraic entropy[END_REF] respectively. Note that in the autonomous case φ n = φ and φ (k) = φ k , and the definition of d coincides with that of the (first) dynamical degree of a dominant rational self-map of P 2 .

Remark 2.11. If instead of P 2 we consider equations as birational mappings of P 1 × P 1 the degrees of iterates are in general different but their rate of growth is the same. Therefore the dynamical degree and algebraic entropy of a mapping (x n , y n ) → (x n+1 , y n+1 ) does not depend on the choice of compactification of C 2 . In fact, it is often more convenient to use P 1 × P 1 for calculation in particular examples as we will do in Sections 3-6, but we will use P 2 in this Section for the general theory.

The kinds of mappings with a space of initial conditions between which full deautonomisation should be shown to distinguish are provided by the following classifications.

2.2.1. Autonomous case. In [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], Diller and Favre classified bimeromorphic self-maps of compact surfaces according to the rate of growth of their spectral radii under iteration. Restating some of their results in the language of birational maps, as studied in discrete integrable systems, yields the following. Proposition 2.12 (Diller-Favre [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). An autonomous birational mapping φ of the plane belongs to one of the following types:

type (1): The degree of φ n is bounded. This type of mapping has a space of initial conditions. type (2): The degree of φ n grows linearly.

This type of mapping does not have a space of initial conditions. type (3): The degree of φ n grows quadratically.

This type of mapping has a space of initial conditions, which is a rational elliptic surface. type (4): The degree of φ n grows exponentially and φ has a space of initial conditions.

The surface X providing the space of initial conditions has rank Pic(X) > 10. type ( 5): The degree of φ n grows exponentially and φ does not have a space of initial conditions.

2.2.2.

Non-autonomous case. If a mapping, autonomous or not, has a space of initial conditions then the dynamical degree is encoded in the induced dynamics on its Picard lattice.

Lemma 2.13 (Takenawa [START_REF] Takenawa | Algebraic entropy and the space of initial values for discrete dynamical systems[END_REF]). If a non-autonomous mapping φ n has a space of initial conditions, the degree of φ (k) is given by

deg φ (k) = Φ k e (0) • e (0) , (2.10)
so the dynamical degree of the mapping is given by the largest eigenvalue of Φ, and the algebraic entropy by its logarithm. Similarly if an autonomous mapping φ has a space of initial conditions then the dynamical degree is given by the largest eigenvalue of φ * .

Using the above fact and the work of Diller-Favre [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], one of the authors obtained the following classification of non-autonomous mappings with a space of initial conditions.

Proposition 2.14 ([24]

). A non-autonomous mapping of the plane with a space of initial conditions belongs to one of the following three types according to the Jordan normal form of Φ: type (a):

Φ ∼    µ 1 . . . µ r+1    ,
where µ i are all roots of unity. In particular Φ l = id for some l > 0 so the degree growth of the mapping is bounded. type (b):

Φ ∼          1 1 1 1 1 µ 1 . . . µ r-2         
, where µ i are all roots of unity. In this case the degree grows quadratically. type (c):

Φ ∼        λ 1 λ µ 1 . . . µ r-1       
, where λ > 1 is a reciprocal quadratic integer or a Salem number, and |µ i | = 1. In this case the degree grows exponentially and the dynamical degree is λ.

We can partially distinguish between the different types above according to rank Pic(X ) as follows Proposition 2.15 ([24]). If rank Pic(X ) ≤ 10, then the mapping belongs to either type (a) or type (b) of Proposition 2.14.

For type (b), when the degree growth is quadratic, one of the authors also showed [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF] that a minimal space of initial conditions is formed of generalised Halphen surfaces as defined by Sakai as follows.

Definition 2.16 (generalised Halphen surface [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], Sakai surface). A rational surface X is called a generalised Halphen surface if it has an effective anticanonical divisor D

∈ | -K X | of canonical type, i.e if D = i m i D i , m i > 0, is its decomposition into irreducible components then [D i ] • K X = 0 for all i.
A generalised Halphen surface has dim | -K X | being equal to either 1 or 0. In the first case X is a rational elliptic surface and in the second case X has a unique anticanonical divisor. The latter is the type of surface which forms the spaces of initial conditions for discrete Painlevé equations, and we call such X a Sakai surface.

From Proposition 2.14 we see that if a non-autonomous mapping has a space of initial conditions and unbounded degree growth, then it must either be a discrete Painlevé equation or non-integrable. So to put the full deautonomisation method on a rigorous footing we need to show that confinement conditions on a sufficiently general deautonomisation distinguish between these two cases.

Remark 2.17. The restriction to basic rational surfaces in the Definitions 2.4 and 2.5 of a space of initial conditions is justified for surfaces associated with mappings of type (b) or (c), since in both cases the surfaces will have infinitely many exceptional curves of the first kind, so by a theorem of Nagata [START_REF] Nagata | On rational surfaces I. Irreducible curves of arithmetic genus 0 or 1[END_REF] they must be basic rational surfaces.

Deautonomisation.

We now define what it means for a non-autonomous mapping to be a deautonomisation of an autonomous one with reference to the space of initial conditions and dynamics on the Picard lattice. Definition 2.18 (deautonomisation). Consider a family of mappings φ n : P 2 P 2 with a space of initial conditions as above. Consider also an autonomous mapping ψ : P 2 P 2 with a space of initial conditions in the sense of Definition 2.1 being a basic rational surface Y with rank Pic(Y ) = rank Pic(X ). Consider a birational morphism (given by a sequence of blow-ups) and denote the geometric basis corresponding to that morphism by (f (0) , . . . , f (r) ). We say that φ n is a deautonomisation of ψ if under the identification κ : Pic(X ) → Pic(Y ),

e (i) → f (i) , (2.11)
the following two conditions hold:

• If F ∈ Pic(X ) represents an effective class ι n (F) in Pic(X n ), then κ(F) ∈ Pic(Y ) is also effective. •
The linear maps Φ and ψ * coincide under the identification, i.e. the following diagram commutes:

Pic(X n ) Pic(X n+1 ) Pic(X ) Pic(X ) Pic(Y ) Pic(Y ). φ * n κ ιn Φ κ ι n+1 ψ * Remark 2.
19. This definition is consistent with that of [START_REF] Carstea | Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations[END_REF] formulated in the context of deautonomising QRT mappings to obtain discrete Painlevé equations. The notion of deautonomisation there involves a choice of fibre from the elliptic fibration of the space of initial conditions for a QRT mapping, which then becomes the unique anticanonical divisor of the surface after allowing locations of centres of blow-ups to move. In particular, a different choice of singular fibre will lead to a different choice of which classes stay effective after deautonomisation, and therefore corresponds to a different way of injecting Pic + (X n ) into Pic + (Y ) via κ • ι -1 n , but this does not change Φ nor the degree growth of the deautonomised mappings.

The crucial fact that we will use for the remainder of the paper is the following Lemma 2.20. If φ n is a deautonomisation of an autonomous mapping in the sense of Definition 2.18, then it has exactly the same degree growth, dynamical degree and algebraic entropy.

2.4.

Period map for rational surfaces with effective anticanonical divisor. In this Subsection, let X be a rational surface with an effective anticanonical divisor

D = i m i D i ∈ | -K X |. Denote the sublattice of Pic(X) spanned by the classes D i = [D i ] of the irreducible components of D by Q = i Z D i . (2.12)
Its orthogonal complement is another sublattice, which we denote

Q ⊥ = {F ∈ Pic(X) | F • D i = 0 ∀ i } , (2.13)
on which the period map will give a C-valued function. When X is a Sakai surface, Q ⊥ is isomorphic to the root lattice of an affine root system and the values of the period map on a basis of simple roots are called the root variables. These appear as n-dependent parameters in discrete Painlevé equations [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF].

Begin by taking a rational 2-form ω on X such that -div ω = D. Let the support of D be D red = i D i , so ω defines a holomorphic symplectic form on X -D red . This gives the mapping χ :

H 2 (X -D red ; Z) → C, Γ → Γ ω.
(2.14)

In order to use this to define a period map on Q ⊥ , we begin with the long exact sequence of relative singular homology for the pair (X, X -D red ), which includes the following:

H 3 (X; Z) → H 3 (X, X -D red ; Z) ∂ * -→ H 2 (X -D red ; Z) i * -→ H 2 (X; Z) j * -→ H 2 (X, X -D red ; Z).
Here i * comes from the natural injection of cycles on X -D red into X, while j * is induced by the quotient of cycles on X by cycles on X -D red . We manipulate this exact sequence using the following facts:

• Lefschetz duality for the pair (X, X -D red ) gives

H k (X, X -D red ; Z) ∼ = H 4-k (D red ; Z). (2.15)
• Poincaré duality for D red gives

H 1 (D red ; Z) ∼ = H 1 (D red ; Z). (2.16)
• Under Poincaré duality for X, we have

H 3 (X; Z) ∼ = H 1 (X; Z) = 0, (2.17) because b 1 = rank H 1 (X; Z) = p+q=1 h p,q = h 1,0 + h 0,1 = 2h 0,1 = 0, since X being rational implies that h 0,1 = 0. • Lefschetz duality as above gives H 2 (X, X -D red ; Z) ∼ = H 2 (D red ; Z) ∼ = Q, so under the
Poincaré duality H 2 (X; Z) ∼ = H 2 (X; Z), in the long exact sequence above we have

ker j * ∼ = Q ⊥ , (2.18)
which is why the orthogonal complement of Q is significant in this construction.

Using these facts, from the sequence above we obtain

0 → H 1 (D red ; Z) → H 2 (X -D red ; Z) → Q ⊥ → 0, (2.19)
which through the mapping χ on H 2 (X -D red ; Z) gives the following.

Definition 2.21 (period map [START_REF] Looijenga | Rational surfaces with an anticanonical cycle[END_REF][START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]). For X with effective anticanonical divisor D and choice of a rational two-form ω on X with -div ω = D, the above construction gives

χ : Q ⊥ → C mod χ(H 1 (D red ; Z)), (2.20)
which we call the period map of X. One can define χ as a C-valued function by making a normalisation of the value of χ on H 1 (D red ; Z) if this is nontrivial.

2.4.1. Computation of the period map. The computation of the period map proceeds along the same lines as in [23, Chapter I, Section 5] and [39, Lemma 21], which we recall here for completeness. The first step in the procedure is guaranteed to be possible for generalised Halphen surfaces, but for surfaces associated with non-integrable mappings does not come automatically either because the existence of an effective anticanonical divisor is not guaranteed or, even if this does exist, it is not obvious how to describe Q ⊥ as a root lattice; we will address this later in this Section.

For an element α ∈ Q ⊥ which can be expressed as the difference of the classes of two exceptional curves of the first kind,

α = [C 1 ] -[C 0 ], (2.21)
the computation of χ(α) is done as follows:

• Find the unique component D k of the anticanonical divisor which intersects both C 1 and C 0 , with multiplicity one, i.e. D k such that

C 1 • D k = C 0 • D k = 1, and C 1 • D j = C 0 • D j = 0 for j ̸ = k. ( 2
χ(α) = 2πi D k ∩C 1 D k ∩C 0 Res D k ω. (2.23)
It is important to note that the integral (2.23) will be computed in coordinates and therefore detects parameters in the mapping, which is fundamental to understanding the mechanism behind the full deautonomisation method.

2.4.2. Dynamics of the values of the period map. The crucial property of the period map for our purposes is how it interacts with the linear dynamics on Pic(X ). Take some linearly independent subset {α i } ⊂ Q ⊥ ⊂ Pic(X ) whose span is closed under Φ, and let M i,j be the matrix representation of Φ with respect to this, i.e.

Φ(α

i ) = j M i,j α j . (2.24)
For each n, take ω n to be a rational 2-form on X n chosen such that φ * n (ω n+1 ) = ω n . Then if we denote by χ n and χ n+1 the period maps defined by ω n and ω n+1 on X n and X n+1 respectively, and the values of the period map of X n by a i (n) = χ n ι -1 n (α i ) , then we have

a i (n + 1) = j M i,j a i (n), (2.25) since χ n+1 ι -1 n+1 (α i ) = χ n φ * n • ι -1 n+1 (α i ) = χ n ι -1 n • Φ(α i ) = j M i,j χ n ι -1 n (α i ) .
The second equality comes from the fact that φ * n (ω n+1 ) = ω n , and that φn is biholomorphic and acts as a change of variables for the integrals.

The fact that the representation in (2.25) is given by exactly the same matrix as in (2.24) provides the sought after bridge between parameter evolution and dynamics on Pic(X ).

2.5. The question of existence of an effective anticanonical divisor. For Sakai surfaces associated with discrete Painlevé equations the existence of an effective anticanonical divisor is built into the definition. However, to carry out the period map construction in the non-integrable case, we must deal with the question of the existence of an effective anticanonical divisor on the surface X n from a space of initial conditions for a mapping of type (c) in Proposition 2.14.

We will need the following fact relating to a space of initial conditions for a non-integrable mapping, the proof of which is found in [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF].

Lemma 2.22 ([24, Lemma 4.15]). For a mapping of type (c) in Proposition 2.14, denote by v ∈ Pic R (X ) the dominant eigenvector corresponding to the eigenvalue λ > 1 of Φ. Then the intersection product is negative definite on v ⊥ ∩ Pic(X ).

Using techniques similar to those in [START_REF] Diller | Rational surface maps with invariant meromorphic two-forms[END_REF] and [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF], we have the following, Theorem 2.23. Let φ n : P 2 P 2 be a non-autonomous, non-integrable mapping which has a space of initial conditions, i.e. a mapping of type (c) in Proposition 2.14. Suppose there exists a sequence of rational 2-forms ω n on P 2 and constants c n ∈ C * such that

φ * n ω n+1 = c n ω n .
(2.26)

Denoting the lift of ω n to X n by ωn = (ρ n • σ -1 n ) * ω = (σ n ) * • (ρ n ) * ω, with its divisor having irreducible decomposition div(ω n ) = i m (n) i D (n) i , (2.27)
then if under the identification of all Pic(X n ) into the Picard lattice Pic(X ) we have, for all n,

ι -1 n (D (n) i ) = ι -1 n+1 (D (n+1) i ), (2.28)
there exists a minimal space of initial conditions formed by surfaces which have effective anticanonical divisors.

Remark 2.24. In particular, if an autonomous mapping ψ : P 2 P 2 is such that ψ * ω = c ω for some rational two-form ω on P 2 and some c ∈ C * , then any deautonomisation of it satisfies the conditions in Theorem 2.23.

Proof of Theorem 2.23. Let the space of initial conditions for the mapping φ n be as in Definition 2.5. Let the decomposition of div(ω n ) on X n into effective and anti-effective parts be div

(ω n ) = D + n -D - n , (2.29)
where D + and D -are effective divisors. Note that -div(ω n ) being effective is equivalent to

D + n = 0.
The idea now is to show that if -div(ω n ) is not effective then the space of initial conditions is not minimal. Indeed suppose D + n ̸ = 0 and lets its decomposition into irreducible components be

D + n = i m i C (n) i , (2.30)
where m i > 0 and C

(n) i are irreducible curves on X n . Denote the classes of these by [C

(n) i ] ∈ Pic(X n ).

Then according to our assumptions the expressions for [C (n)

i ] in terms of the geometric basis for Pic(X n ) corresponding to π n do not depend on n and we have the following well-defined elements of Pic(X ):

D + i = ι -1 n ([C (n) i ]), D + = i m i ι -1 n ([D] + i ).
(2.31)

We will show that there exists some D + i that corresponds to the class of an exceptional curve of the first kind and which is periodic under Φ, and that its orbit gives a collection of classes of exceptional curves of the first kind which are mutually disjoint, so by Lemma 2.9 the space of initial conditions is not minimal.

The fact that φ * n preserves the two-form ω n in the sense of (2.26) implies that φ * n ωn+1 = c n ωn , and φ * n (div(ω n+1 )) = div(ω n ). Further, since φn is an isomorphism, for any irreducible curve C on X n we have ord(C, div(ω n )) = ord( φn (C), div(ω n+1 )), (2.32) which allows us to deduce that Φ must permute the D + i in a way that preserves their multiplicity in the canonical class. In particular there exists a permutation σ such that Φ(D + i ) = D + σ(i) , so all D + i are in the part of Pic(X ) periodic under Φ. Denoting by ℓ the order of the permutation σ, we have

D + i • v = Φ ℓ (D + i ) • Φ ℓ v = λ ℓ D + i • v , (2.33)
and since λ > 1 we deduce that D + i ∈ v ⊥ ∩Pic(X ) for every i, so the intersection form is negative definite on the sublattice span Z D + i ⊂ Pic(X ). This allows us to deduce that at least one of the D + i must correspond to an exceptional curve of the first kind, as follows. Taking the intersection of D + with the canonical class in Pic(X ) we have

D + • K X = D + • D + -D + • D -≤ D + • D + < 0,
where the last inequality comes from the intersection form being negative definite on the span of the D + i . Since D + corresponds to an effective class we then have that some

D + i satisfies D + i • K X < 0. Since D + i ∈ v ⊥ ∩ Pic(X) we also deduce D + i • D + i < 0.
The genus formula then allows us to deduce that D + i must correspond to an exceptional curve of the first kind. It remains to be shown that the elements of Pic(X ) in the orbit of this D + i under Φ are mutually orthogonal. Letting k ∈ Z be such that Φ k D + i = D + j ̸ = D + i , we again use the negative definiteness of the intersection form on the span of the D + i to show that

0 > D + i + Φ k D + i 2 = -2 + 2D + i • Φ k D + i =⇒ 1 > D + i • Φ k D + i ≥ 0, (2.34)
so D + i • Φ k D + i = 0 and the orbit of D + i gives a collection of mutually disjoint exceptional curves of the first kind which are permutated by the mapping.

Finally we note that after a minimisation, performed by blowing down these curves, we still have a space of initial conditions for the same φ n in the sense of Definition 2.5, and we can apply the above procedure as many times as is necessary to obtain a minimal space of initial conditions formed of anticanonical surfaces. Note that the procedure is guaranteed to terminate since a space of initial conditions for a non-integrable mapping must have rank Pic(X ) > 10 according to Proposition 2.15. □ 2.6. Bases for the orthogonal sublattice. The next problem which must be addressed is ensuring that on surfaces associated with non-integrable mappings, one can find appropriate elements of Q ⊥ on which to compute the period map. For the remainder of this Section we assume all surfaces have effective anticanonical divisors.

Definition 2.25 (root basis of Q ⊥ ). Suppose a Z-basis {α i } for Q ⊥ is such that each α i can be expressed as the difference of two classes of exceptional curves of the first kind

α i = [C 1 i ] -[C 0 i ] and the numbers c i,j = 2 α i •α j α i •α i form
the entries of a generalised Cartan matrix as defined in [START_REF] Kac | Infinite-dimensional Lie algebras[END_REF]. Then we call this a root basis for Q ⊥ . Following [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], we will refer to the values of the period map on a root basis as root variables.

For each type of generalised Halphen surface in his classification, Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF] found a root basis with a corresponding generalised Cartan matrix that is of affine type. In [START_REF] Takenawa | Algebraic entropy and the space of initial values for discrete dynamical systems[END_REF], Takenawa constructed the space of initial conditions for the Hietarinta-Viallet mapping [START_REF] Hietarina | Singularity confinement and chaos in discrete systems[END_REF], which has an effective anticanonical divisor. Takenawa was also able to find a root basis in this case, corresponding to the generalised Cartan matrix

  2 -2 -2 -2 2 -2 -2 -2 2   , (2.35)
which is of hyperbolic type H

71 [START_REF] Wan | Introduction to Kac-Moody algebra[END_REF]. Takenawa also constructed an example with root basis corresponding to the matrix

  2 -3 -3 -3 2 -3 -3 -3 2   , (2.36)
which is of neither finite, affine, nor hyperbolic type.

For non-integrable mappings it is not immediately clear that we can always find a root basis, but for the purpose of full-deautonomisation it will be sufficient to show that one can always construct a linearly independent subset of Q ⊥ that is closed under Φ and whose span includes the dominant eigenvector. To this end we define the following. Definition 2.26 (sufficient subset of Q ⊥ ). For a space of initial conditions constructed from a mapping with unbounded degree growth, suppose a linearly independent subset {β j } of Q ⊥ is such that the following two conditions hold:

• Each β j can be expressed as the difference of two classes of exceptional curves of the first kind C 1 j and C 0 j , i.e.

ι n (β j ) = [C 1 j ] -[C 0 j ], • Φ preserves span Z {β j }.
If additionally the following requirement is met, which depends on whether the mapping is of type (b) or (c) of Proposition 2.14, then we call {β j } a sufficient subset:

• type (c) : The dominant eigenvector v ∈ Pic R (X ) of Φ is contained in span R {β j },
• type (b) : Some element in Pic R (X ) which is a generalised eigenvector for the eigenvalue 1 of Φ, but not an eigenvector, is contained in span R {β j }.

For non-autonomous mappings of type (b) of Proposition 2.14, it was shown in [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF] that they correspond to Sakai surfaces, for which we can always find a sufficient subset in Q ⊥ . However, in order to show that we can always find a sufficient subset for mappings of type (c) in Proposition 2.14 we will require the following Lemma consisting of several results from [START_REF] Mase | Singularity confinement as an integrability criterion[END_REF].

Lemma 2.27 ([26]). Denote the periodic part of Pic(X ) under Φ by

P Φ = {F ∈ Pic(X ) | ∃ m ∈ Z \ {0} such that Φ m (F) = F} , (2.37)
and let µ Φ (t) ∈ Q[t] denote the minimal polynomial of Φ as a linear transformation of Pic(X )/ P Φ . If there exists F ∈ Pic(X )/ P Φ , F ̸ = 0, such that a polynomial

ψ(t) ∈ Q[t] satisfies ψ[Φ]F = 0 mod P Φ , (2.38)
then ψ(t) shares a factor with µ Φ (t). For mappings of type (c) in Proposition 2.14, µ Φ (t) is precisely the minimal polynomial of λ, the dominant eigenvalue of Φ, over Q. We denote this minimal polynomial as µ λ (t).

We will also require the following result, which follows from the classification of the anticanonical divisors for spaces of initial conditions for non-integrable mappings (Theorem A.17) that is given in Appendix A.

Proposition 2.28. Consider a mapping φ n which is non-integrable, i.e. of type (c) in Proposition 2.14, with a minimal space of initial conditions provided by surfaces X n which are anticanonical, i.e. X n has an effective anticanonical divisor D = j m j D j . Then, there exists j such that

• m j = 1, • X n has an exceptional curve of the first kind C such that C • D i = δ i,j .
We are now ready to prove the following. Proposition 2.29. For a minimal space of initial conditions for a non-integrable mapping, i.e. of type (c) in Proposition 2.14, one can always find a sufficient subset of Q ⊥ .

Proof. Let the surfaces forming the space of initial conditions be X n , with D = j m j D j as the effective anticanonical divisor of X n . Then let [C] ∈ Pic(X n ) be the class of the exceptional curve C of the first kind as in Proposition 2.28, which intersects exactly one component D j of the anticanonical divisor D = j m j D j . The element of D k = ι -1 n [D k ] ∈ Pic(X ) corresponding to this class must lie in P Φ , so let ℓ ∈ Z ≥0 be the smallest nonnegative integer such that Φ ℓ (D k ) = D k . Furthermore, let C = ι -1 n [C] be the element of Pic(X ) corresponding to the exceptional curve of the first kind C, which cannot lie in P Φ because of the minimality of the space of initial conditions. Then we can construct a sufficient subset as follows. First let

β 0 = (Φ ℓ -1)C, (2.39)
which by construction corresponds to the difference of two classes of exceptional curves of the first kind. Then let β 1 = Φβ 0 , β 2 = Φβ 1 and so on until the process terminates and some β h+1 is a linear combination of β 0 , . . . , β h , say Φβ h + h i=0 a i β i = 0. Then {β 0 , . . . , β h } by construction satisfies the first two conditions in Definition 2.26, so it remains only to show that v ∈ span R {β 0 , . . . , β h }. From the construction of this subset we have that

ψ[Φ](Φ ℓ -1)C = 0, (2.40)
where

ψ(t) = (t h+1 + h i=0 a i t i ) ∈ Q[t].
Since we are in the third case of Proposition 2.14, Lemma 2.27 implies that ψ has µ λ (t), the minimal polynomial over Q of the dominant eigenvalue of Φ, as a factor. Therefore the characteristic polynomial of the restriction of Φ to span R {β 0 , . . . , β h } has µ λ (t) as a factor, so the dominant eigenvector v is contained in span R {β 0 , . . . , β h } as required. □ Definition 2.30 (sufficient deautonomisation). Consider a deautonomisation φ n of an autonomous mapping ψ in the sense of Definition 2.18, with space of initial conditions given by surfaces X n . For a sufficient subset {β 1 , . . . , β r } of Q ⊥ , consider the Z-linear map

χ : span Z {β 1 , . . . , β r } → (C mod χ(H 1 (D red ; Z))) r , β → χ Xn (ι n (β)) , (2.41)
where χ Xn is the period map of X n . If χ is injective, then we call the deautonomisation sufficient. This means that the linear system for the evolution with n, of the values of χ on the sufficient subset, will have the same characteristic polynomial as that of Φ restricted to span Z {β 1 , . . . , β r }.

In the case of type (c) mappings this characteristic polynomial will have (t -λ) as a factor (where λ is the dominant eigenvalue of Φ), whereas for type (b) mappings it has (t -1) 2 as a factor.

The construction of a sufficient deautonomisation depends significantly on the form in coordinates of the autonomous mapping. For mappings that take the form of mappings belonging to classes I-VI in the QRT-type classification, we give procedures for constructing sufficient deautonomisations in Appendix B. If a deautonomisation is sufficient in the above sense, this means that the evolution of values of the period map on the sufficient subset is guaranteed to give a characteristic polynomial with the dynamical degree of the mapping as a root, and we have the following main result of the paper. Theorem 2.31. If an autonomous mapping has a space of initial conditions and unbounded degree growth, and preserves a rational two-form, it has a minimal space of initial conditions formed of anticanonical surfaces. Then the period map parametrisation of the coefficients in a sufficient deautonomisation gives a linear system of difference equations for their evolution, the characteristic polynomial of which has the dynamical degree of the mapping as a root.

A family of mappings of QRT Class III form

The first example we will illustrate in detail deals with a family of mappings that are of the form of mappings in QRT class III, i.e.

(x n+1 + x n ) (x n + x n-1 ) = f (x n ), (3.1)
where f is rational. From this point on we will make use of notation such as

x n-1 = x, x n = x, x n+1 = x, (3.2)
to denote up-and down-shifts in n of both variables and parameters. We consider the family of equations

(x + x) (x + x) = m i=0 (x 2 -a 2 i ) m j=1 (x 2 -b 2 j ) , (3.3)
where a i ∈ C\{0}, i = 0, . . . m and b j ∈ C\{0}, j = 1, . . . m, for m ∈ Z ≥1 . Singularities of the mapping correspond to zeroes of the numerator or denominator of the rational function on the right-hand side of (3.3), and calculating along the same lines as in the examples in Section 1 we find that all of these singularities are confined and we have the singularity patterns {±a i , ∓a i } , and {±b j , ∞, ∓b j } .

(3.4)

3.1. Full deautonomisation by singularity confinement. In this case it turns out that the following deautonomisation suffices to detect the dynamical degree of the mapping

(x + x) (x + x) = m i=0 (x -a i ) (x -d i ) m j=1 (x -b j ) (x -c j ) . (3.5)
Here a i , b j , c j , d i are now n-dependent and will be required to evolve such that the structure of confined singularities persists. We shall now derive these evolutions explicitly. First consider the singularities that correspond to zeroes of the numerator of the right-hand side of (3.5). In order to have the same kind of confinement behaviour of these singularities as the patterns {±a i , ∓a i } in the autonomous case, we require singularities which appear when some iterate x takes a value for which the numerator of the right-hand side vanishes, to lead to a next iterate x that is a zero of the numerator of the up-shifted version of the equation. Considering first the singularity x = a k , for some k ∈ {0, . . . , m} with x = u free, we introduce a small parameter ε and compute

x = u, x = a k + ε, x = -a k + O(ε), (3.6)
and similarly for the singularity

x = d k , x = u, x = d k + ε, x = -d k + O(ε). (3.7)
For these singularities to be confined in the same way after deautonomisation we impose that x = a k leads to x = dk , and x = d k leads to x = āk , which requires the confinement conditions dk = -a k , āk = -d k ,

for which it can be verified by direct calculation that we indeed have the confined singularity patterns {a k , dk } and {d k , āk }. This is tantamount to requiring that the lines blown down by the mapping ( x, x) → (x, x) are sent to indeterminacies of the next iteration (x, x) → (x, x).

For the singularities x = b k and x = c k we have

x = u, x = b k + ε, x = i (b k -a i )(b k -d i ) (b k + u) j̸ =k (b k -b j ) j (b k -c j ) ε -1 + O(1), x = -b k + m j=1 ( bj + cj ) + m i=0 (ā i + di ) + O(ε), (3.9)
and a similar evolution for x = c k + ε obtained from (3.9) by interchanging the roles of b k and c k . We require that these singularities are confined in the same way as in the autonomous case, namely through x taking a value which is a root of the denominator of the right-hand side in the twice up-shifted version of (3.5), corresponding to an indeterminacy (∞, bj ) or (∞, cj ) of the mapping (x, x) → ( x, x). Which of these indetermacies the singularities x = b k , x = c k are confined through can be chosen, without loss of generality, to correspond to the patterns {b k , ∞, ck } and {c k , ∞, bk } via the conditions

ck = -b k + m j=1 ( bj + cj ) + m i=0 (ā i + di ), bk = -c k + m j=1 ( bj + cj ) + m i=0 (ā i + di ).
(3.10)

For the purpose of full deautonomisation in this case it is sufficient (as explained in Appendix B) to take the parameters in the numerator of the right-hand side of (3.5) to be constant, after which the conditions (3.8) require

a i = -d i ∈ C\{0}, for all n. (3.11)
Then the remaining confinement conditions (3.10) give a linear system for the evolution of

b = (b 1 , . . . , b m ) T , c = (c 1 , . . . , c m ) T :     b c b c     =     1 m 1 m 0 -I m 1 m 1 m -I m 0 I m 0 0 0 0 I m 0 0         b c b c     , (3.12)
where I m is the m × m identity matrix, and 1 m is the m × m matrix with all entries being equal to one. The matrix on the right-hand side is similar under column permutations to

M = 1 2m -I 2m I 2m 0 , (3.13)
so the characteristic polynomial of the linear system (3.12) coincides, up to cyclotomic factors, with that of M , which can be computed as a block determinant to be

det(M -t I 4m ) = t 2 + 1 2m-1 (t 2 -2mt + 1). (3.14)
Therefore we see that the characteristic polynomial has as a root

λ = m + m 2 -1, (3.15)
which by the results of Section 2 we can conclude to be the dynamical degree of the original equation (3.3), as we will go on to demonstrate in detail.

3.2.

Space of initial conditions. We now illustrate how the mechanism by which the dynamical degrees of the family of mappings (3.3) were obtained above, fits into the framework of Section 2, beginning with the construction of the space of initial conditions for the autonomous mapping. We take the mapping initially on P 1 × P 1 (see Remark 2.11) and perform a sequence of blowups to regularise it as an automorphism. We do this in the usual way by introducing X = 1/x, Y = 1/y so P 1 × P 1 is covered by the four affine charts (x, y), (X, y), (x, Y ) and (X, Y ) and the equation (3.3) defines via y = x the mapping

φ : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = -x + m i=0 (x 2 -a 2 i ) (x + y) m j=1 (x 2 -b 2 j ) , ȳ = x. (3.16)
In order to regularise this as an automorphism we require blowups of points given in coordinates by

a ± i : (x, y) = (±a i , ∓a i ), b ± i : (x, y) = (∞, ±b i ), b± i : (x, y) = (±b i , ∞), (3.17)
after which we have the rational surface X as shown in Figure 4. Denoting the composition of the blow-ups by π : X → P 1 × P 1 and the exceptional divisors by

y = ∞ x = ∞ x + y = 0 b+ 1 b- 1 • • • b+ m b- m b + 1 b - 1 . . . b - m b + m a + 0 a - 0 a + m a - m . . . P 1 × P 1 π D 2 D 3 D 1 B+ 1 B- 1 • • • B+ m B- m B + 1 B - 1 . . . B + m B - m A + 0 A - 0 A + m A - m . . . X
A ± i = π -1 (a ± i ), B ± j = π -1 (b ± j ), B± j = π -1 ( b± j ), we have Pic(X) = ZH x + ZH y + m i=0 (ZA + i + ZA - i ) + m j=1 (ZB + j + ZB - j + Z B+ j + Z B- j ), (3.18)
where given by x + y = 0, y = ∞, and x = ∞ respectively. This is the pole divisor of the two-form on X given in the original coordinates by dx∧dy x+y , which is preserved by the mapping. Through calculations in charts, we find that the mapping acts by pullback on Pic(X) as follows.

H x = π * (O P 1 (1) × 1) and H y = π * (1 × O P 1 ( 1
φ * :        H x → (2m + 1)H x + H y - m i=0 A + i - m i=0 A - i - m j=1 B+ j - m j=1 B- j , H y → H x , A ± i → H x -A ∓ i , B ± j → H x -B± j , B± j → B ∓ j . (3.19)
While it is possible to compute the characteristic polynomial of the matrix of φ * with respect to this basis for Pic(X), it will be more convenient to do this in terms of the root basis we will introduce after deautonomisation.

Space of initial conditions for deautonomised version.

In what follows we will use a straightforward adaptation of the formulation of the space of initial conditions for a nonautonomous mapping of Section 2 to account for families of surfaces obtained by blow-ups from P 1 × P 1 , rather than P 2 (see Remark 2.11). In particular the notions of geometric bases for Pic(X n ) and identification of Picard groups into a single Z-module via ι n : Pic(X ) → Pic(X n ) are defined when X n is obtained by blow-ups from P 1 × P 1 in the natural way.

Consider the deautonomised equation (3.5) defining the mapping

φ n : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = -x + m i=0 (x -a i )(x -d i ) (x + y) m j=1 (x -b j )(x -c j ) , ȳ = x, (3.20)
subject to the confinement conditions derived above, namely

āi = -d i , di = -a i , bk = -ck + m j=1 (b j + c j ) + m i=0 (a i + d i ), ck = -bk + m j=1 (b j + c j ) + m i=0 (a i + d i ). (3.21)
To construct the space of initial conditions we introduce the surface X n obtained by blowing up P 1 × P 1 at the following points, as shown in Figure 5:

a i : (x, y) = (a i , -a i ), e i : (x, y) = (d i , -d i ), i = 0, . . . , m, b j : (x, y) = (∞, bj ), c j : (x, y) = (∞, cj ), bj : (x, y) = (b j , ∞), cj : (x, y) = (c j , ∞), j = 1, . . . , m. (3.22)
Denoting the composition of the blow-ups by π n : X n → P 1 × P 1 , and the exceptional divisors by

A i = π -1 n (a i ), E i = π -1 n (e i ), B j = π -1 n (b j ), C j = π -1 n (c j ), Bj = π -1 n ( bj ), Cj = π -1 n (c j ), we identify all Pic(X n ) into a single Z-module Pic(X ) = ZH x + ZH y + m i=0 (ZA i + ZE i ) + m j=1 (ZB j + ZC j + Z Bj + Z Cj ), (3.23)
where

ι n (H x ) = π * n (O P 1 (1) × 1), ι n (A i ) = [A i
] and so on. The intersection product gives the symmetric bilinear form on Pic(X ) defined by

H x • H y = 1, H x • H x = H y • H y = 0,
with all generators that correspond to exceptional divisors of blow-ups being of self-intersection -1 and orthogonal to all other generators. Each surface X n has an effective anticanonical divisor given by D = D 1 + D 2 + D 3 ∈ | -K Xn |, where D 1 , D 2 , D 3 are the proper transforms of the curves given by x + y = 0, y = ∞, and x = ∞ respectively.

y = ∞ x = ∞ x + y = 0 b1 c1 • • • bm cm b 1 c 1 . . . c m b m a 0 e 0
a m e m . . .

P 1 × P 1 π n D 2 D 3 D 1 B1 C1 • • • Bm Cm B 1 C 1 . . . C m B m A 0 E 0 A m E m . . . X n
= π -1 n+1 • φ n • π n : X n → X n+1
, and its pullback induces the following lattice automorphism

Φ = ι -1 n • φ * n • ι n+1 of Pic(X ): Φ :              H x → (2m + 1)H x + H y - m i=0 A i - m i=0 E i - m j=1 Bj - m j=1 Cj , H y → H x , A i → H x -E i , E i → H x -A i , B j → H x -Bj , C j → H x -Cj , Bj → C j , Cj → B j .
(3.24)

Remark 3.2. This means that the mapping (3.5) with the confinement conditions (3.21) is a deautonomisation of (3.3) in the sense of Definition 2.18, via κ : Pic(X ) → Pic(X),

A i → A + i , E i → A - i , B j → B + j , C j → B - j , Bj → B+ j , Cj → B- j .
(3.25)

3.4. Root basis and period map. With the space of initial conditions and linear transformation Φ in hand, we now calculate the period map of X n and confirm the mechanism by which the dynamical degree appeared in the confinement conditions. We take the rational two-form to be that given in the initial affine charts for P 1 × P 1 by

ω = k dx ∧ dy x + y = -k dX ∧ dy X(1 + Xy) = -k dx ∧ dY Y (xY + 1) = k dX ∧ dY XY (X + Y ) , (3.26)
with k ∈ C * being arbitrary at this stage, and denote its lift to X n under π n by ωn = π * n ω.

(3.27)

We have -div(ω n ) = D 1 +D 2 +D 3 , and the elements of Pic(X ) corresponding to the irreducible components are

D 1 = H x +H y - m i=0 A i - m i=0 E i , D 2 = H y - m j=1 Bj - m j=1 Cj , D 3 = H x - m j=1 B j - m j=1
C j , (3.28) so we denote their span and its orthogonal complement respectively by

Q = span Z {D 1 , D 2 , D 3 } , Q ⊥ = {F ∈ Pic(X ) | F • D i = 0} . (3.29)
To find a root basis for Q ⊥ in the sense of Definition 2.25, we note that when m = 1, X n becomes a Sakai surface of (additive) type A

(1) * 2 in Sakai's classification scheme so we can use the root basis there, which is formed of simple roots for an affine root system of type E

(1) 6 (see [39, Appendix A] or [22, 8.2.16]). It turns out that this choice can be generalised to the m > 1 case as follows.

Proposition 3.3. We have a root basis for Q ⊥ ⊂ Pic(X ) given by

Q ⊥ = 2m i=0 Zα i + 2m-1 j=1 Zβ j + 2m-1 j=1 Zγ j + Zε 1 + Zε 2 , (3.30)
where

ε 1 = H y -A 0 -B 1 , ε 2 = H x -A 0 -B1 , α 0 = A 0 -E 0 , α 1 = E 0 -A 1 , α 2 = A 1 -E 1 , . . . α 2m-1 = E m-1 -A m , α 2m = A m -E m , β 1 = B 1 -C 1 , β 2 = C 1 -B 2 , β 3 = B 2 -C 2 , . . . β 2m-2 = C m-1 -B m , β 2m-1 = B m -C m , γ 1 = B1 -C1 , γ 2 = C1 -B2 , γ 3 = B2 -C2 , . . . γ 2m-2 = Cm-1 -Bm , γ 2m-1 = Bm -Cm . (3.31)
Proof. In this case K X ̸ ∈ Q ⊥ and we have no overlap between Q and Q ⊥ , so since rank Pic(X ) = 6m + 4 we need to find 6m + 1 elements to construct a root basis. We begin by taking the obvious choices of differences of pairs of classes of exceptional divisors intersecting the components D 1 , D 2 , D 3 , which provides 6m -1 elements and leads to α 0 , . . . , α 2m , β 1 , . . . , β 2m-1 and γ 1 , . . . , γ 2m-1 as above. For the additional two elements required to form a root basis, we take cues from the known choice in the m = 1 case and set ε 1 to be the difference of the class H y -A 0 of the proper transform of the line y = -a 0 and the class B 1 of B 1 , which both intersect D 3 . Similarly we choose ε 2 to be the difference of the class H x -A 0 of the proper transform of the line x = a 0 and the class B1 of B1 , which both intersect D 2 . The fact that these choices lead to a root basis in the sense of Definition 2.25 is immediate, and in Figure 6 we present the graph corresponding to the generalised Cartan matrix given by the intersection numbers of the elements, i.e. nodes corresponding to elements, with edges joining them if the pair of elements has intersection 1. We remark that this graph already appeared in the work of Looijenga [START_REF] Looijenga | Rational surfaces with an anticanonical cycle[END_REF] as the intersection configuration of a root basis for a surface with an anticanonical cycle of length three -this root basis will appear again in a way more directly related to Looijenga's work in Section 4. □ We now compute the period mapping on this basis via the procedure outlined in Section 2. Note that in this case D red is the union of three copies of P 1 meeting at a single point so H 1 (D red ; Z) = 0, and the only normalisation we need in order to define χ as a C-valued function is a choice of the constant k appearing in ω as in (3.26). A convenient choice leads to the following, in which the period map is that on X n and we slightly abuse notation by writing α i etc. when more precisely we mean their corresponding elements ι n (α i ) ∈ Pic(X n ).

β 2m-1 . . . β 1 ε 1 α 0 ε 2 γ 1 . . . γ 2m-1 α 1 α 2 . . .
Lemma 3.4. The root variables for the root basis in Proposition 3.3 are given by

χ(α 0 ) = a 0 -d 0 , χ(α 1 ) = d 0 -a 1 , χ(α 2 ) = a 1 -d 1 , χ(α 3 ) = d 1 -a 2 , . . . χ(α 2m-1 ) = d m-1 -a m , χ(α 2m ) = a m -d m , χ(ε 1 ) = -a 0 -b1 , χ(β 1 ) = b1 -c1 , χ(β 2 ) = c1 -b2 , χ(β 3 ) = b2 -b2 , . . . χ(β 2m-2 ) = cm-1 -bm , χ(β 2m-1 ) = bm -cm , χ(ε 2 ) = -a 0 + b 1 , χ(γ 1 ) = -b 1 + c 1 , χ(γ 2 ) = -c 1 + b 2 , χ(γ 3 ) = -b 2 + c 2 , . . . χ(γ 2m-2 ) = -c m-1 + b m , χ(γ 2m-1 ) = -b m + c m .
(3.32)

Proof. In order to compute the period mapping using the procedure outlined in Section 2, we first note that the residues of ωn along the components of the anticanonical divisor D are given in coordinates by

Res D 1 ωn = Res s=0 k dx ∧ ds s = -kdx, Res D 2 ωn = Res Y =0 -k dx ∧ dY Y (xY + 1) = kdx,
Res D 3 ωn = Res X=0 -k dX ∧ dy X(1 + Xy) = -kdy, (3.33)
where s = x + y. To calculate χ(α i ) for i = 0, . . . , 2m we note that we can express

A j -E k = [A j ] -[E k ]
as the difference of two exceptional curves of the first kind which intersect D 1 , which is the proper transform of the line x + y = 0. Therefore using the residue computed above we have

χ(A j -E k ) = 2πi A j ∩D 1 E k ∩D 1 Res D 1 ωn = 2πi x=a j x=d k -kdx = -2πik(a j -d k ). (3.34)
Similarly we find

χ( Bj -Ck ) = 2πi Bj ∩D 2 Ck ∩D 2 Res D 2 ωn = 2πi x=b j x=c k kdx = 2πik(b j -c k ), χ(B j -C k ) = 2πi B j ∩D 3 C k ∩D 3 Res D 3 ωn = 2πi x= bj y= ck -kdy = -2πik( bj -ck ), (3.35)
which allows us to deduce the values of χ on all elements of the root basis except for ε 1 and ε 2 , but these are computed similarly. For example for ε 1 , we use its expression as the difference of classes H y -A 0 and B 1 of exceptional curves intersecting D 3 , and note that H y -A 0 corresponds to the proper transform of the line y = -a 0 so we compute

χ(ε 1 ) = 2πi y=-a 0 y= b1 -kdy = 2πik(a 0 + b1 ), (3.36)
and ε 2 is dealt with similarly. Finally we choose k such that 2πik = -1, and we have the result. □

With this lemma in hand, we can compute how the values of the period map on the root basis evolve with n as dictated by the confinement conditions (3.21). Denoting vectors of elements of the root basis by

α e = (α 0 , α 2 , . . . , α 2m ) T , α o = (α 1 , α 3 , . . . , α 2m-1 ) T , β o = (β 1 , β 3 , . . . , β 2m-1 ) T , β e = (β 2 , β 4 , . . . , β 2m ) T , γ o = (γ 1 , γ 3 , . . . , γ 2m-1
) T , and γ e = (γ 2 , γ 4 , . . . , γ 2m ) T , the following is obtained by direct calculation.

Proposition 3.5. The evolution with n of the root variables for the surface X n is given by

χ = M Φ • χ, (3.37)
where χ = (χ(ε 1 ), χ(ε 2 ), χ(α o ), χ(α e ), χ(β e ), χ(β o ), χ(γ e ), χ(γ o ) ) T , χ = χ(ε 1 ), χ(ε 2 ), χ( ᾱo ), χ( ᾱe ), χ( βe ), χ( βo ), χ(γ e ), χ(γ o ) T ,

and

M Φ =             0 -1 * * * * * * 1 2m * * * * * * 0 0 -I m+1 * 0 0 0 0 0 0 0 I m 0 0 0 0 0 0 0 0 0 0 -I m 0 0 0 0 0 0 0 0 -I m 0 0 0 0 I m * 0 0 0 0 0 0 0 I m 0 0             (3.38)
and where χ, χ are the period maps of the surfaces X n , X n+1 respectively, and bars on root basis elements indicate their counterparts in Pic(X n+1 ) under the identification ι n+1 .

As expected, the evolution of the root variables is given by exactly the same matrix as that of Φ on Q ⊥ with respect to the root basis. Indeed, by direct computation using the expressions in Proposition 3.3 for the root basis in terms of generators of Pic(X ) and Φ as it appears in Proposition 3.1, we see that the evolution of the root basis for Q ⊥ under Φ takes the form

Φ : δ → M Φ δ, (3.39)
where δ = (ε 1 , ε 2 , α o , α e , β e , β o , γ e , γ o ) T . Further, it can be verified by direct calculation that all elements of the root basis aside from ε 1 and ε 2 are in the periodic part P Φ of Pic(X ) under Φ, so the minimal polynomial with the dynamical degree of the mapping as a root can be obtained as follows.

Proposition 3.6. The transformation of Q ⊥ / P Φ , induced by Φ is given with respect to the basis {ε 1 + P Φ , ε 2 + P Φ } by the matrix

0 -1 1 2m , (3.40)
whose characteristic polynomial is (t 2 -2mt + 1). In particular the dynamical degree of the mapping is equal to the largest root λ = m + √ m 2 -1 of this polynomial.

Remark 3.7. The fact that, in order to read off the dynamical degree of the mapping from the confinement conditions (3.21), it was sufficient to keep a i and d i constant as in (3.11) is precisely because the root variables that are fixed because of this choice, correspond to elements of Q ⊥ ∩ P Φ . Keeping these parameters fixed still gives a sufficient deautonomisation in the sense of Definition 2.30.

A family of mappings of QRT Class IV form

We next consider the family of mappings of the form

(xx -1) (x x -1) = m i=0 (x -a i )(x -a -1 i ) m j=1 (x -b j )(x -b -1 j ) , (4.1)
where a i ∈ C\{0}, i = 0, . . . m and b j ∈ C\{0}, j = 1, . . . m, for m ∈ Z ≥1 . Similarly to the example in Section 3, we have singularities corresponding to zeroes of the numerator and denominator of the rational function on the right-hand side of (4.1), and these are confined with singularity patterns a ±1 i ,

Full deautonomisation by singularity confinement. To determine the dynamical degree of the mapping (4.1), we take only a sufficient deautonomisation (see Appendix B) given by

(xx -z) (x x -z) = m i=0 (x -a i )(x -d i ) m j=1 (x -b j )(x -c j ) , (4.2)
where the n-dependence of z, a i , d i is to be determined, and the rest of the parameters on the right-hand side are taken to be constant:

bj = b j , cj = c j . (4.3)
For the singularities x = b j , and x = c j to be confined in the same way as in the autonomous mapping (4.1) we can choose to require patterns {b j , ∞, c j } and {c j , ∞, b j }, which for constant b j and c j leads to

c j = b -1 j , for j = 1, . . . , m. (4.4)
Hence these patterns are exactly the same as in the autonomous case, i.e. b ±1 j , ∞, b ∓1 j . We enforce confinement of the singularities x = a i and x = d i respectively through the patterns {a i , di } and {d i , āi }, by imposing the conditions

āi = z d i , di = z a i . (4.5)
Finally we note that to properly deautonomise equation (4.1), we must ensure x = 0 is not a singularity. To see that this is the case before deautonomisation, note that at x = 0 the right-hand side of (4.1) is equal to 1, so when recast as a mapping (4.10) a cancellation occurs so that there is no factor of x in the denominator of the rational function giving x. To preserve this property under deautonomisation we require

z z = m i=0 a i d i , (4.6)
so in particular from this condition and its up-shifted version we have zz =

m i=0 āi di = m i=0 (zd -1 i )(za -1 i ) = z 2m+1 z , (4.7)
from which we deduce the equation governing the n-dependence of z to be zz = z 2m . (

Note that this constraint is multiplicative, as opposed to the additive constraint found for the mappings in Section 3. However, upon linearisation it leads to the same characteristic polynomial as in Section 3:

t 2 -2mt + 1, (4.9)
the largest root of which is again λ = m + √ m 2 -1, which we will confirm to be the dynamical degree of the mapping (4.1). 4.2. Space of initial conditions. We now construct the space of initial conditions for the autonomous equation (4.1), recast as the mapping

φ : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = 1 x + m i=0 (x -a i )(x -a -1 i ) x(xy -1) m j=1 (x -b j )(x -b -1 j ) , ȳ = x, (4.10)
and we will recycle notation for blow-up points, exceptional divisors, etc. from Section 3. For this mapping we blow up the points

a ± i : (x, y) = (a ±1 i , a ∓1 i ), b ± i : (x, y) = (∞, b ±1 i ), b± i : (x, y) = (b ±1 i , ∞), (4.11)
after which we have the rational surface X as shown in Figure 7. This surface has effective anticanonical divisor

D = D 1 + D 2 + D 3 ∈ | -K X |
, where D 1 , D 2 , D 3 are the proper transforms of the curves given by xy = 1, y = ∞, and x = ∞ respectively. This is the pole divisor of the two-form on X given in the original coordinates by dx∧dy xy-1 , which is preserved by the mapping.

y = ∞ x = ∞ xy = 1 b+ 1 b- 1 • • • b+ m b- m b + 1 b - 1 . . . b - m b + m a + 0 a - 0 a + m a - m . . . P 1 × P 1 π D 2 D 3 D 1 B+ 1 B- 1 • • • B+ m B- m B + 1 B - 1 . . . B + m B - m A + 0 A - 0 A + m A - m . . . X Figure 7
. Space of initial conditions for Class IV example (autonomous).

Through calculations in charts, we find that the mapping acts by pullback on Pic(X) in exactly the same way as the example from Class III studied in Section 3, as given by (3.19). While this immediately implies that the dynamical degree must also be the same, the way in which the confinement conditions must be linearised before the characteristic polynomial for Φ can be read off in this case is different. This is related to the geometry of the anticanonical divisor D as we will show below. 4.3. Space of initial conditions for deautonomised version. Consider the deautonomised equation (4.2), with confinement conditions as derived above, defining the mapping

φ n : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = z x + m i=0 (x -a i )(x -d i ) x(xy -z) m j=1 (x -b j )(x -b -1 j ) , ȳ = x, (4.12)
with parameters satisfying bi

= b i , āi = z d i , di = z a i , z z = m i=0 a i d i . (4.13)
We construct the surface X n by blowing up P 1 × P 1 at the following points, as shown in Figure 8:

a i :(x, y) = (a i , za -1 i ), e i : (x, y) = (d i , zd -1 i ), i = 0, . . . , m b ± j :(x, y) = (∞, b ±1 j ), b± j : (x, y) = (b ±1 j , ∞), j = 1, . . . , m. (4.14)
We identify all Picard groups of the surfaces X n into the single lattice Pic(X ) as before and obtain the following.

Proposition 4.1. With the confinement conditions (4.13), the mapping φ n in (4.12) becomes an isomorphism

φn = π -1 n+1 • φ n • π n : X n → X n+1
, and its pullback induces the following lattice automorphism

Φ = ι -1 n • φ * n • ι n+1 of Pic(X ): Φ :        H x → (2m + 1)H x + H y - m i=0 A i - m i=0 E i - m j=1 B+ j - m j=1 B- j , H y → H x , A i → H x -E i , E i → H x -A i , B ± j → H x -B± j , B± j → B ∓ j . (4.15)
y = ∞ x = ∞ xy = z b+ 1 b- 1 • • • b+ m b- m b + 1 b - 1 . . . b - m b + m a 0 e 0
a m e m . . .

P 1 × P 1 π n D 2 D 3 D 1 B+ 1 B- 1 • • • B+ m B- m B + 1 B - 1 . . . B + m B - m A 0 E 0 A m E m . . . X n Figure 8
. Space of initial conditions for Class IV example (deautonomised).

4.4. Root basis and period map. Again we calculate the period map of X n and confirm the mechanism by which the dynamical degree appeared in the confinement conditions, including the linearisation that was required to obtain a difference equation from which the minimal polynomial for the dynamical degree could be read off. The rational two-form preserved by the mapping in this case is given in charts for P 1 × P 1 by

ω n = k dx ∧ dy xy -z = -k dX ∧ dy X(y -X z) = -k dx ∧ dY Y (x -Y z) = k dX ∧ dY XY (1 -XY z) , (4.16)
with k ∈ C * , and we again denote its lift to X n under π n by ωn = π * n ω n . (4.17)

Direct calculation shows that the two-form is preserved by the mapping, i.e.,

dx ∧ dȳ xȳ -z = dx ∧ dy xy -z , (4.18)
and even though ω n on P 1 × P 1 depends on n, we are still within the remit of Theorem 2.23. Indeed X n has effective anticanonical divisor -div(ω n ) = D 1 + D 2 + D 3 , and the elements of Pic(X ) corresponding to the irreducible components are given by

D 1 = H x + H y - m i=0 A i - m i=0 E i , D 2 = H y - m j=1 B+ j - m j=1 B- j , D 3 = H x - m j=1 B + j - m j=1 B - j .
(4.19) Defining the subsets Q, Q ⊥ ⊂ Pic(X ) as above, we find the same root basis as in Section 3, but changes in notation mean that B j and C j are replaced with B + j and B - j respectively, and their tilded versions similarly, i.e.

ε 1 = H y -A 0 -B + 1 , ε 2 = H x -A 0 -B+ 1 , α 0 = A 0 -E 0 , α 1 = E 0 -A 1 , α 2 = A 1 -E 1 , . . . α 2m-1 = E m-1 -A m , α 2m = A m -E m , β 1 = B + 1 -B - 1 , β 2 = B - 1 -B + 2 , β 3 = B + 2 -B - 2 , . . . β 2m-2 = B - m-1 -B + m , β 2m-1 = B + m -B - m , γ 1 = B+ 1 -B- 1 , γ 2 = B- 1 -B+ 2 , γ 3 = B+ 2 -B- 2 , . . . γ 2m-2 = B- m-1 -B+ m , γ 2m-1 = B+ m -B- m . (4.20)
In particular the generalised Cartan matrix for this basis is the same as in Section 3 encoded by the graph in Figure 6. The key difference between this example and that of the previous section is the parametrisation by the period map of the coefficients in the mapping and the centres of blow-ups, because of the fact that the anticanonical divisor is a cycle and H 1 (D red ; Z) ∼ = Z. In particular the period map is defined by the surface only modulo χ(H 1 (D red ; Z)) ∼ = Z, which can be seen in terms of the residues of ωn along the irreducible components,

Res D 1 ωn = Res s=0 k dx ∧ ds xs = -k dx x ,
Res D 2 ωn = Res Y =0 -k dx ∧ dY Y (x -Y z) = k dx x ,
Res D 3 ωn = Res X=0 -k dX ∧ dy X(y -X z) = -k dy y , (4.21)
where s = xyz. Note that evaluating integrals of these residues will lead to logarithmic functions of parameters in the mapping. Computing along the same lines as in the proof of Lemma 3.4 and imposing appropriate normalisations of both k and the value χ(ℓ) for a generator ℓ of H 1 (D red ; Z) as in [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], we find the following.

Lemma 4.2. The root variables of X n for the root basis above are related to the parameters in the mapping (4.12) according to

e χ(α 0 ) = a 0 d -1 0 , e χ(α 1 ) = d 0 a -1 1 , e χ(α 2 ) = a 1 d -1 1 , e χ(α 3 ) = d 1 a -1 2 , . . . e χ(α 2m-1 ) = d m-1 a -1 m , e χ(α 2m ) = a m d -1 m , e χ(ε 1 ) = za -1 0 b -1 1 , e χ(β 1 ) = b 2 1 e χ(β 2 ) = b -1 1 b -1 2 , e χ(β 3 ) = b 2 2 , . . . e χ(β 2m-2 ) = b -1 m-1 b -1 m , e χ(β 2m-1 ) = b 2 m , e χ(ε 2 ) = a -1 0 b 1 , e χ(γ 1 ) = b -2 1 , e χ(γ 2 ) = b 1 b 2 , e χ(γ 3 ) = b -2 2 , . . . e χ(γ 2m-2 ) = b m-1 b m , e χ(γ 2m-1 ) = b -2 m . (4.22)
The evolution of the root variables is again given by the matrix of Φ with respect to the root basis, and we can also explain how the evolution of the parameter z gave the scalar difference equation (4.8) from which the dynamical degree could be read off after linearisation. Expressing z in terms of the root variables of X n we have z = e χ(ε 1 )-χ(ε 2 )+χ(β 1 ) , (4.23) and in light of this we introduce

v 1 = ε 1 -ε 2 + β 1 , v 2 = Φ(v 1 ), (4.24)
and see that

Φ(v 2 ) = -v 1 + 2mv 2 + G, (4.25)
where

G = 2β 1 -2m-1 j=2 (2m -j)β j + 2γ 1 -2m-1 j=2 (2m -j)γ j .
Noting that all β j , γ j are in the intersection of Q ⊥ with the periodic part P Φ of Pic(X ) under Φ, we can verify the matrix of the induced transformation of Q ⊥ / P Φ with respect to the basis {v 1 + P Φ , v 2 + P Φ } to be 0

1 -1 2m , (4.26)
so again we can read off the dynamical degree from its characteristic polynomial. The fact that this characteristic polynomial appeared so neatly from the evolution of z is because in our chosen deautonomisation the parameters b j are kept constant, which leads to

e χ(G) = 1, (4.27)
and therefore, through the period map, equation (4.25) becomes

z = e χ(v 2 ) = e -χ(v 1 )+2mχ(v 2 )+χ(G) = z 2m z-1 , (4.28)
which is precisely the confinement condition from which we read off the dynamical degree.

A family of mappings of QRT Class I form

The next example we will consider is the family of equations (1.12) introduced in the Introduction, namely

x + x = 1 x 2m , (5.1)
where m ∈ Z ≥1 , which are of the form of Class I QRT-type mappings. This mapping admits the confined singularity pattern {0, ∞ 2m , 0}, and if we take the deautonomisation

.x + x = 1 x 2m - b x , (5.2)
the n-dependence of b required for the singularity confinement to persist is b -2mb + b = 0, from which we can read off minimal polynomial t 2 -2mt + 1 for the dynamical degree of the mapping (5.1), namely λ = m + m 2 -1.

(5.3) 5.1. Space of initial conditions. The space of initial conditions for equation (5.1) was constructed in [START_REF] Mase | Singularity confinement as an integrability criterion[END_REF], which we recall here with some added details which will be necessary for deautonomisation and the computation of the period map. Consider the mapping

φ : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = 1 -yx 2m x 2m , ȳ = x, (5.4)
defined by equation (5.1). In contrast to the examples studied above, here, to construct the space of initial conditions we will have to blow up infinitely near points, i.e. those lying on the exceptional divisors of previous blow-ups. For this reason we will need to introduce notation for charts covering exceptional divisors according to the following convention: after blowing up a point p i given in some affine chart (x, y) by

p i : (x, y) = (x * , y * ), (5.5)
the exceptional divisor E i of the blowup of p i is covered by two affine coordinate charts (u i , v i ) and (U i , V i ) given by

x = u i v i + x * , y = v i + y * , u i = x -x * y -y * , v i = y -y * , and
x = V i + x * , y = U i V i + y * , U i = y -y * x -x * , V i = x -x * . (5.6)
In particular the exceptional divisor E i has in these charts the local equations v i = 0 and V i = 0.

For the case at hand we require 8m blow-ups in total, 4m each over the points (x, y) = (0, ∞) and (x, y) = (∞, 0) which we outline now. Firstly letting p 1 : (x, y) = (0, ∞) and introducing coordinates for the exceptional divisor as above we require first 2m blowups of points p k+1 : (U k , V k ) = (0, 0) for k = 1, . . . 2m -1. Note that each of these points p k+1 lies on the proper transform of the line y = ∞ under the blowups of p 1 , . . . , p k . Then we find the point p 2m+1 : (U 2m , V 2m ) = (1, 0) away from the proper transform of y = ∞, and after blowing this up we require another sequence of blow-ups of points p 2m+2 , . . . , p 4m given in coordinates by p 2m+k+1 : (u k , v k ) = (0, 0), k = 1, . . . , 2m -1, though these points are away from the proper transforms of y = ∞ or any of the previous exceptional divisors. Similarly for p1 : (x, y) = (∞, 0) we initially blow up 2m points on the proper transform of the line x = ∞ given by pk+1 : (ũ k , ṽk ) = (0, 0), where we have used tildes to denote the coordinates introduced in blowing up points pi to distinguish them from those coming from blow-ups of p i . Then similarly we find p2m+1 : (ũ 2m , ṽ2m ) = (1, 0) followed by a sequence p2m+2 , . . . , p4m given by p2m+k+1 : (ũ k , ṽk ) = (0, 0), k = 1, . . . , 2m -1. The resulting surface is shown in Figure 9, where E i and Ẽi are exceptional divisors from the blow-ups of p i and pi respectively, and we have introduced notation D i , Di for the curves which will be shown to be the components of an anticanonical divisor of X whose classes in Pic(X) are

D 0 = H y - 2m i=1 E i , D i = E i -E i+1 , i = 1, . . . , 4m -1, D0 = H x - 2m i=1
Ẽi , Di = Ẽi -Ẽi+1 , i = 1, . . . , 4m -1.

(5.7)

The fact that the mapping becomes an automorphism of X is verified by calculation in charts. We take the deautonomised mapping (5.2) as φ n :

y = ∞ x = 0 x = ∞ y = 0 p1 p2 • • • p4m p 1 p 2 • • • p 4m P 1 × P 1
P 1 × P 1 → P 1 × P 1 , (x, y) → (x, ȳ), x = 1 -bx 2m-1 -yx 2m x 2m , ȳ = x, (5.8)
with the confinement condition b -2mb + b = 0.

(5.9) The points to be blown up to construct the surface X n that provides a space of initial conditions for mapping (5.8), are given by the same expressions in coordinates as for the autonomous case with the exception of the final points p 4m and p4m , which for the non-autonomous mapping are p 4m : (u 4m-1 , v 4m-1 ) = (b, 0), p4m : (ũ 4m-1 , ṽ4m-1 ) = ( b, 0).

(5.10)

Denoting again the exceptional divisors arising from the blow-ups by E i = π -1 n (p i ) and Ẽi = π -1 n ( pi ), we identify all Pic(X n ) into the single Z-module

Pic(X ) = ZH x + ZH y + 4m i=1 ZE i + 4m i=1 Z Ẽi , (5.11)
where [E i ] = ι n (E i ) and [ Ẽi ] = ι n ( Ẽi ).

Proposition 5.1. With the confinement condition (5.9), the mapping φ n in (5.8) becomes an isomorphism φn = π -1 n+1 • φ n • π n : X n → X n+1 , and its pullback induces the following lattice automorphism

Φ = ι -1 n • φ * n • ι n+1 of Pic(X ): Φ : H x → 2mH x + H y - 4m i=1 E i , H y → H x , E i → Ẽi , Ẽi → H x -E 4m+1-i .
(5.12) 5.3. Root basis and period map. The rational two-form ω on P 1 × P 1 preserved by φ n is given by

ω = kdx ∧ dy = -k dx ∧ dY Y 2 = -k dX ∧ dy X 2 = k dX ∧ dY X 2 Y 2 , (5.13)
where k ∈ C * , and we denote its lift to X n by ωn = π * n ω. The pole divisor of ωn is effective and is given in terms of the curves D j , Dj by

-div ωn = 2D 0 + 2m j=1 jD j + 2m-1 j=1 (2m -j)D 2m+j + 2 D0 + 2m j=1 j Dj + 2m-1 j=1 (2m -j) D2m+j . (5.14)
To construct a root basis in this case we perform a similar procedure to that outlined in the proof of Proposition 2.29. The singularity pattern {0, ∞ 2m , 0} corresponds to the movement of the following exceptional curves of the first kind under the mapping according to

[x = 0] -→ Ẽ4m -→ E 4m -→ [y = 0], (5.15)
where [x = 0] and [y = 0] indicate proper transforms under the blowups. Passing to Pic(X ) we have

H x -E 1 Φ ←----Ẽ4m Φ ←----E 4m Φ ←----H y -Ẽ1 , (5.16)
so we take H y -Ẽ1 and see that the component of D it intersects corresponds to D1 = Ẽ1 -Ẽ2 , which has period 4 under Φ, so we compute

Φ 4 (H y -Ẽ1 ) = 2mH x + H y - 4m i=1 E i -Ẽ1 .
(5.17)

and obtain an element of Q ⊥ given by

α 1 = (Φ 4 -1)(H y -Ẽ1 ) = 2mH x - 4m i=1 E i . (5.18)
Applying Φ to α 1 we see that a root basis can be found as follows.

Proposition 5.2. We have a root basis for Q ⊥ given by

α 1 = 2mH x - 4m i=1 E i , α 2 = 2mH y - 4m i=1 Ẽi . (5.19)
In Figure 10 we give the intersection diagram of the irreducible components of the anticanonical divisor as well as the diagram corresponding to the generalised Cartan matrix formed by the numbers c i,j = 2

α i •α j α i •α i , which here is 2 -2m -2m 2 .
(5.20)

In particular in the m = 1 case X n is a Sakai surface of type D

7 , and the the root basis in Proposition 5.2 is a basis of simple roots for an affine root system of type A

(1) 1 |α| 2 =4
, where the non-standard length of roots does not affect the fact that the matrix with entries c i,j = 2

α i •α j α i •α i becomes the usual generalised Cartan matrix of type A (1) 1 . D 2m D 0 D0 D2m D2m+1 . . . D4m-1 D2m-1 . . . D1 D 2m+1 . . . D 4m-1 D 2m-1 . . . D 1 α 1 α 2 2m
Figure 10. Intersection diagrams of anticanonical divisor components and root basis for Class I example

The evolution of the root basis under Φ is given by Φ :

α 1 α 2 → 0 -1 1 2m α 1 α 2 , (5.21)
and we see the factor (t 2 -2mt + 1) from the characteristic polynomial of Φ on Pic(X ) as the characteristic polynomial of its restriction to Q ⊥ , and we can deduce that the dynamical degree of the mapping is λ = m + √ m 2 -1. To see this on the level of the confinement condition on b we use the expressions for the root basis as differences of classes of exceptional curves of the first kind and find the root variables for the root basis in Proposition 5.2 to be

χ(α 1 ) = b, χ(α 2 ) = -b. (5.22)
Therefore the confinement condition b = 2mbb corresponds to evolution of the root variables according to the same matrix as Φ on the root basis as in (5.21), i.e.

χ(ᾱ 1 ) = b = -χ(α 2 ), χ(ᾱ 2 ) = -b = b -2mb = χ(α 1 ) + 2mχ(α 2 ), (5.23)
and we see how the parameter evolution reflects the dynamical degree.

6. A family of Class V-type mappings with short patterns

The last example we will study in detail is one to which other integrability tests based on singularity analysis cannot be applied directly. Consider the autonomous equation

x + x -2z x + x x + x -2z x + x = (x -z) 2 -a 2 1 (x -z) 2 -a 2 2 (x 2 -b 2 1 )(x 2 -b 2 2 ) , (6.1)
where the parameters z, a i , b j are constants. This mapping has the singularity confinement property, with singularity patterns

{z ± a i , z ∓ a i }, {±b j , ∓b j }, (6.2)
for i, j = 1, 2. Because of the absence of singularity patterns of length ≥ 3, neither Halburd's method [START_REF] Halburd | Elementary calculations of degree growth and entropy for discrete equations[END_REF] nor its 'express' version [START_REF] Ramani | Calculating algebraic entropies: an express method[END_REF] can be used to compute the dynamical degree of this mapping -see [37, Section 5] for an explanation of why this is the case. However full deautonomisation is still effective, as we will demonstrate.

6.1. Full deautonomisation by singularity confinement. To illustrate full deautonomisation here we will consider the family of non-autonomous mappings

x + x -2z x + x x + x -2 z x + x = m i=1 (x -z -a i )(x -z -d i ) m j=1 (x 2 -b 2 j ) , (6.3)
for some integer m ≥ 1, where z, a i and d i are allowed to evolve with n but b j are kept constant: bj = b j . (6.4)

The family of mappings (6.3) contains a sufficient deautonomisation of the autonomous equation (6.1) when m = 2. We will show that full deautonomisation can be used to determine the dynamical degrees of the whole family of non-autonomous mappings (6.3), which in particular recovers that of the autonomous case (6.1). With the parameters b j constant we automatically have the singularity patterns {±b j , ∓b j }, but in order for the remaining singularities to be confined similarly to the autonomous case we must require the confinement conditions

z + āi = z -d i , z + di = z -a i , (6.5)
under which the mapping admits the singularity patterns {z + a i , z + di } and {z + d i , z + āi }.

To completely preserve the singularity structure and obtain a deautonomisation of the mapping in the sense of Definition 2.18 we must also ensure that x = ∞ is not a singularity, just as in the case of the autonomous mapping (6.1). This requires, at each n the constraint

2z + 2 z - m i=1 (z + a i ) - m i=1 (z + d i ) = 0. (6.6)
Similarly to in Section 4 we combine this condition and its up-shift to find z -2(m -1)z + z = 0, (6.7) from which we read off the characteristic polynomial t 2 -2(m -1)t + 1 with largest root

λ = m -1 + m(m -2), (6.8)
which is equal to the dynamical degree of the mapping when m ≥ 2. When m = 1 the mapping is periodic, with period 4, and therefore has bounded degree growth.

6.2. Space of initial conditions for non-autonomous mapping. The non-autonomous equation (6.3) defines the mapping

φ n : P 1 × P 1 P 1 × P 1 , (x, y) → (x, ȳ), x = -x - 2z(x + y -2 z)g(x) (x + y)(f (x) -g(x)) + 2 zg(x) , ȳ = x, where f (x) = m i=1 (x -z -a i ) (x -z -d i ) , g(x) = m j=1 x 2 -b 2 j , (6.9)
subject to the confinement conditions derived previously:

z + āi = z -d i , z + di = z -a i , 2z + 2 z - m i=1 (z + a i ) - m i=1 (z + d i ) = 0. (6.10)
The points to be blown up to construct the space of initial conditions are given by a i : (x, y) = ( z + a i , z -a i ), e i : (x, y) = ( z + d i , z -d i ) b ± j : (x, y) = (±b j , ∓b j ), (6.11) and we denote the exceptional divisors by A i = π -1 (a i ), E i = π -1 (e i ), and

B ± i = π -1 (b ± i ).
The surface X n has an effective anticanonical divisor D = D 1 + D 2 , where D 1 and D 2 are the two rational curves given in coordinates by

D 1 : x + y = 0, D 2 : x + y -z = 0, (6.12)
which intersect at a single point (x, y) = (∞, ∞) with multiplicity two. These correspond to the elements of Pic(X )

D 1 = H x + H y - m i=1 B + i - m i=1 B - i , D 2 = H x + H y - m i=1 A i - m i=1 E i , (6.13)
where

A i = ι n ([A i ]), E i = ι n ([E i ]), and
B ± i = ι n ([B ± i ])
; we give a schematic representation of X n in Figure 11.

y = ∞ x = ∞ b + 1 b - 1 b + 2 b - 2 b + m b - m . . . a 1 e 1 a 2 e 2 a m e m P 1 × P 1 π n D 1 B + 1 B - 1 B + 2 B - 2 B + m B - m D 2 A 1 E 1 A 2 E 2 A m E m . . . . . . X n Figure 11
. Space of initial conditions for Class V example (deautonomised). Proposition 6.1. With the confinement conditions (6.10), the mapping φ n in (6.9) becomes an isomorphism φn = π -1 n+1 • φ n • π n : X n → X n+1 , and its pullback induces the following lattice automorphism

Φ = ι -1 n • φ * n • ι n+1 of Pic(X ): Φ :        H x → 2mH x + H y - m i=1 (A i + E i ) - m j=1 (B + j + B - j ), H y → H x , A i → H x -E i , E i → H x -A i , B ± i → H x -B ∓ i . (6.14)
The characteristic polynomial of Φ as written in Proposition 6.1 can be verified to have a factor of t 2 -2(m -1)t + 1, meaning that the dynamical degree of the mapping is equal to 1 if m = 1 or m = 2, and otherwise is equal to the largest root λ > 1 of this polynomial as in (6.8). We remark this family of mappings contains all three types from the classification in Proposition 2.14. When m = 2 the autonomous mapping preserves an elliptic fibration, with its deautonomisation being a discrete Painlevé equation of (additive) surface type A

(1) * 1 [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], and when m = 1 the degree growth of the equation is bounded. 6.3. Root basis and period map. The non-autonomous mapping preserves the n-dependent two-form ω n on P 1 × P 1 given by

ω n = k z dx ∧ dy (x + y)(x + y -z) , (6.15)
where k ∈ C * , since under the mapping φ n given by (6.9) we have

kz dx ∧ dȳ (x + ȳ)(x + ȳ -z) = k z dx ∧ dy (x + y)(x + y -z) . (6.16)
The anticanonical divisor in this case is then

-div ωn = D = D 1 + D 2 , (6.17)
and the fact that D 1 and D 2 intersect at a single point with multiplicity two means that H 1 (D red ; Z) = 0, which is reflected in the parametrisation of the coefficients by root variables being additive. Proposition 6.2. We have a root basis for Q ⊥ given by

Q ⊥ = 4m-1 i=0 Zα i , (6.18)
where

α 0 = H x -H y , α 1 = E 1 -E 2 , α 2 = E 2 -E 3 , . . . α m-1 = E m-1 -E m , α m = E m -A 1 , α m+1 = A 1 -A 2 , α m+2 = A 2 -A 3 , . . . α 2m-1 = A m-1 -A m , α 2m = H y -A m -B + 1 , α 2m+1 = B + 1 -B + 2 , α 2m+2 = B + 2 -B + 3 , . . . α 3m-1 = B + m-1 -B + m , α 3m = B + m -B - 1 , α 3m+1 = B - 1 -B - 2 , α 3m+2 = B - 2 -B - 3 , . . . α 4m-1 = B - m-1 -B - m . (6.19)
We give the intersection diagram of this root basis in Figure 12, which when m = 2 reduces to the Dynkin diagram associated with the symmetry type E 12 also appeared in the work of Looijenga [START_REF] Looijenga | Rational surfaces with an anticanonical cycle[END_REF] through the root basis for a rational surface with an anticanonical cycle of length two, i.e. two rational curves intersecting at two distinct points each with multiplicity one. In a way similar to how the examples in Sections 3 and 4 share the same intersection diagram of their root bases but have additive and multiplicative natures of their root variables due to differing ranks of H 1 (D red ; Z), the example in the present section has a multiplicative counterpart. Explicitly the family of non-autonomous mappings is given by

xx -z 2 xx -1 x x -z 2 x x -1 = m i=1 (xz -1 -a i )(xz -1 -a -1 i ) m j=1 (x -b j )(x -b -1 j ) , (6.20)
and it has the form of Class VI mappings in the QRT-type classification. The confinement conditions, space of initial conditions and period map for this multiplicative counterpart of the present example can be worked out along the same lines as in the other sections. where α = (α 1 , . . . , α 2m-1 , α 2m+1 , . . . , α 4m-1 ) T and U is a known square matrix of size 4m -2 satisfying U 2 = I 4m-2 . We therefore see the factor t 2 -2(m -1)t + 1 in the characteristic polynomial of Φ as expected and confirm the dynamical degree of the original mapping and its deautonomised version as read off from the confinement conditions.

To compute the period map we note that the residues of ωn along the components of the anticanonical divisor are given in coordinates by where s 1 = x + y and s 2 = x + yz. Therefore, with k chosen for convenience to be 1, the root variables of X n for the root basis in Proposition 6.2 are related to the parameters in the mapping (6.9) as follows, and the evolution of these root variables with the confinement conditions can be checked directly to match with the action of Φ on the root basis. The parameter z that satisfies the recurrence whose characteristic polynomial encodes the dynamical degree corresponds to the root variable χ(α 0 ). Computing the action of Φ on α 0 we find (Φ 2 -2(m -1)Φ + 1)α 0 = 0, (6.24) which through the period mapping gives the confinement condition on z, and we see again how the dynamical degree appears in the confinement conditions.

Conclusion

In summary, in this paper we have provided an explanation (as summarised in Theorem 1.1) of the correspondence between confinement conditions for a vast class of birational mappings of the plane and their dynamical degree. As we explained, this correspondence underlies the efficacy of the method of full deautonomisation by singularity confinement, as an integrability test for such mappings. While the problem of effectiveness of the anticanonical divisor class means that the method is not guaranteed to work for birational mappings in general, examples with a space of initial conditions but no effective anticanonical divisor are rare, and Theorem 2.23 can be regarded as an affirmative answer to a non-autonomous counterpart to the conjecture initially posed by Gizatullin, in the case of mappings preserving a rational two-form.

Another interpretation of the results in this paper is as a generalisation of Takenawa's attempt [START_REF] Takenawa | Discrete dynamical systems associated with root systems of indefinite type[END_REF] towards a version of the Sakai theory of discrete Painlevé equations for non-integrable mappings, involving rational surfaces associated via Q ⊥ to root systems of indefinite type. In particular non-integrable mappings from the class to which Theorem 2.23 applies provide a suite of new examples with indefinite type root bases, extending some which appeared in Looijenga's study of rational surfaces with anticanonical cycle [START_REF] Looijenga | Rational surfaces with an anticanonical cycle[END_REF]. While we have not confirmed this, it is natural to expect that the symmetry groups of the families of surfaces constructed in this paper can be described in terms of the Weyl group associated to Q ⊥ . Further, while a classification of the kinds of effective anticanonical divisors of surfaces forming the space of initial conditions for non-integrable mappings cannot be given by a finite list as in the discrete Painlevé case, we have given a local classification of the intersection configuration of irreducible components in Appendix A. q q q q q q q q q q q q q D 1 D N D N -1 where z, w, A and α 1 , . . , α M are allowed to evolve with n, but β 1 , . . . , β N can be taken to be constant.

  • [D] : the linear equivalence class of D ∈ Div(X). We will generally use calligraphic script to indicate classes of divisors, i.e. D = [D].

. 22 )

 22 This unique component is guaranteed to exist by Riemann-Roch, Serre duality and the genus formula, as explained in [23, Chapter I, Section 0]. • Denote the points of intersection of these curves with D k as D k ∩ C 1 and D k ∩ C 0 . The value of χ(α) is then computed using the residue formula (see [23, Chapter I, Section 5] for details) as

Figure 4 .

 4 Figure 4. Space of initial conditions for Class III example (autonomous).

  )) correspond to classes of lines of constant x and y respectively, and we have used calligraphic script to indicate classes of exceptional divisors. Note that, as guaranteed by Theorem 2.23, the surface X has effective anticanonical divisor given by D = D 1 + D 2 + D 3 , where D 1 , D 2 , D 3 are the proper transforms of the curves

Figure 5 .Proposition 3 . 1 .

 531 Figure 5. Space of initial conditions for Class III example (deautonomised).

α 2m Figure 6 .

 2m6 Figure 6. Intersection diagram of root basis for Q ⊥ in Class III example

Figure 9 .

 9 Figure 9. Space of initial conditions for Class I example

  corresponding discrete Painlevé equation. Remark 6.3. The diagram in Figure

α 1 α 2 .Figure 12 .

 212 Figure 12. Intersection diagram of root basis for Q ⊥ in Class V example

Res D 1

 1 ωn = Res s 1 =0 k z dx ∧ ds 1 s 1 (s 1z) = kdx,Res D 2 ωn = Res s 2 =0 k z dx ∧ ds 2 (s 2 + z)s 2 = -kdx,(6.22)

χ(α 0

 0 ) = 2 z, χ(α 1 ) = -d 1 + d 2 , χ(α 2 ) = -d 2 + d 3 , . . . χ(α m-1 ) = -d m-1 + d m , χ(α m ) = -d m + a 1 , χ(α m+1 ) = -a 1 + a 2 , χ(α m+2 ) = -a 2 + a 3 , . . . χ(α 2m-1 ) = -a m-1 + a m , χ(α 2m ) =z + a m -b 1 , χ(α 2m+1 ) = b 1 -b 2 , χ(α 2m+2 ) = b 2 -b 3 , . . . χ(α 3m-1 ) = b m-1 -b m , χ(α 3m ) = b m + b 1 , χ(α 3m+1 ) = -b 1 + b 2 , χ(α 3m+2 ) = -b 2 + b 3 , . . . χ(α 4m-1 ) = -b m-1 + b m .(6.23)

Proof. 1 D 2 D 3 ( 2 )

 1232 By Lemma A.11, we have2 = D 1 • D 2 ≤ i̸ =1 m i D 1 • D i = 2, which implies that D 2 • D i = 0 for all i ̸ = 1, 2. Replacing D 2 with D 1 , we have D 1 • D i = 0 for all i ̸ = 1, 2. Therefore, D 1 + D 2 is a connected component of D. Since D is connected, D must coincide with D 1 + D 2 . □ Lemma A.13. If D 1 • D 2 =1 and m 1 = m 2 = 1, then only the following types of configurations are possible: (1) D = D 1 + D 2 + D 3 and the configuration has only one intersection: N ≥ 3 and D = D 1 + • • • + D N forms a cycle of P 1 (after reordering the indices):

2 m 8 ) 13 )B. 4 .B. 5 .B. 6 .

 2813456 i D i , which implies that D has only one component D ℓ other than D 1 , D 2 such that D 2 • m ℓ D ℓ = 1. In particular, m ℓ = 1 and D 2 • D ℓ = 1. We may assume that ℓ = 3. If D 1 , D 2 , D 3 intersect at one point, then none of D 1 , D 2 , D 3 intersects D -D 1 -D 2 -D 3 . Since D is connected, D must be D 1 + D 2 + D 3 in this case. B.2. Class II. Consider the three-point equation xx = f (x), (B.6)where f rational. This falls into the remit of Theorem 2.23, since when recast as a mapping (x, y) → (x, ȳ),Let f be of the formf (x) = x k P M (x) Q N (x) , (B.9)where k ∈ Z and P M and Q N are polynomials of degree M and N respectively with no factors of x. A sufficient deautonomisation is given by takingf (x) = Ax k (x -α 1 )(x -α 2 ) . . . (x -α M ) (x -β 1 )(x -β 2 ) . . . (x -β N ) , (B.10)where all α 1 , . . . , α M , β 1 , . . . , β N and A are allowed to evolve with n.B.3. Class III. Consider the three-point equation(x + x)(x + x) = f (x), (B.11)where f is rational. This falls into the remit of Theorem 2.23, since when recast as a mapping (x, y) → (x, ȳ), Let f be of the formf (x) = P M (x) Q N (x) , (B.14)where P M and Q N are polynomials of degree M and N respectively. A sufficient deautonomisation is given by takingf (x) = A (x -α 1 )(x -α 2 ) . . . (x -α M ) (x -β 1 )(x -β 2 ) . . . (x -β N ) , (B.15)where β 1 , . . . , β N are allowed to evolve with n, but α 1 , . . . , α M and A can be taken to be constant. Class IV. Consider the three-point equation(xx -1)(x x -1) = f (x), (B.16)where f is rational. This falls into the remit of Theorem 2.23, since when recast as a mapping (x, y) → (x, ȳ), preserves the rational two-formdx ∧ dȳ xȳ -1 = dx ∧ dy xy -1 . (B.18)Let f be of the formf (x) = P M (x) Q N (x) (B.19)where P M and Q N are polynomials of degree M and N respectively. A sufficient deautonomisation is given by(xx -z)(x x -w) = A (x -α 1 )(x -α 2 ) . . . (x -α M ) (x -β 1 )(x -β 2 ) . . . (x -β N ) , (B.20)where z, w, A and α 1 , . . . , α M are allowed to evolve with n, but β 1 , . . . , β N can be taken to be constant. Class V. Consider the three-point equationx + x -1 x + x x + x -1 x + x = f (x), (B.21)where f is rational. This falls into the remit of Theorem 2.23, since when recast as a mapping (x, y) → (x, ȳ), x + x -1x+ x = x + y x + y -1 f (x), ȳ = x, (B.22)it preserves the rational two-form dx ∧ dȳ(x + ȳ -1)(x + ȳ) = dx ∧ dy (x + y -1)(x + y) . (B.23)Let f be of the formf (x) = P M (x) Q N (x) ,(B.24)where P M and Q N are polynomials of degree M and N respectively. A sufficient deautonomisation is given byx + x -z x + x x + x -w x + x = A (x -α 1 )(x -α 2 ) . . . (x -α M ) (x -β 1 )(x -β 2 ) . . . (x -β N ) , (B.25)where z, w, A and α 1 , . . . , α M are allowed to evolve with n, but β 1 , . . . , β N can be taken to be constant. Class VI. Consider the three-point equationxx -z 2 xx -1 x x -z 2 x x -1 = f (x), (B.26)where f is rational and z is constant. This falls into the remit of Theorem 2.23, since when recast as a mapping (x, y)→ (x, ȳ), xx -z 2 xx -1 = xy -1 xy -z 2 f (x), ȳ = x,(B.27) it preserves the rational two-form dx ∧ dȳ (xȳ -z 2 )(xȳ -1) = dx ∧ dy (xy -z 2 )(xy -1) . (B.28) Let f be of the form f (x) = P M (x) Q N (x) , (B.29) where P M and Q N are polynomials of degree M and N respectively. A sufficient deautonomisation is given by xx -z xx -1 x x -w x x -1 = A (x -α 1 )(x -α 2 ) . . . (x -α M ) (x -β 1 )(x -β 2 ) . . . (x -β N ) , (B.30)

Acknowledgements. AS is supported by a Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Research in Japan and also acknowledges the support of JSPS KAK-ENHI Grant Numbers 21F21775 and 22KF0073. TM and RW would also like to acknowledge support from JSPS, respectively through JSPS grant numbers 18K13438 and 23K12996, and 22H01130.

Conflict of interest statement. The authors have no conflicts of interest to declare. Data availability statement. There is no associated data.

Appendix A. Classification of anticanonical divisors for confining non-integrable mappings

In this Appendix we provide a classification of anticanonical divisors (Theorem A.17) for spaces of initial conditions that correspond to non-integrable mappings, i.e. mappings with dynamical degree greater than 1. As mentioned in [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF], the theory of spaces of initial conditions for mappings of the plane allows for two description that are, however, essentially the same. While the description introduced in the main part of this paper is more intuitive, the other one, by a rational surface and a Cremona isometry, is more compatible with the general theory needed in such a classification. Therefore, in this appendix, we shall use the description by Cremona isometries. The following are our assumptions and notations that we shall use in this appendix:

• X is a rational surface.

• σ is a Cremona isometry. That is, σ is a Z-linear transformation of Pic(X) that satisfies -σK X = K X , -(σF 1 ) • (σF 2 ) = F 1 • F 2 for any F 1 , F 2 ∈ Pic(X), if F ∈ Pic(X) is effective, then so is σF . • λ > 1: the maximum eigenvalue of σ, which is irrational.

• v ∈ Pic R (X): the dominant eigenvector of σ, which we may assume is nef.

• X is minimal as a space of initial conditions.

• The anticanonical class -K X is effective. That is, we have

Proof. The proof of [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]Proposition 6] is still valid in our case.

Proof. Take a geometric basis H, E 1 , . . . , E K ∈ Pic(X) and let π : X → P 2 be the corresponding blow-down. Using this basis, each [D j ] is expressed as

Here,

From here on, we fix j. If c j0 = 0, then D j is the strict transform of some exceptional curve of π. Therefore, D j is rational and g a (D j ) = 0. Let us consider the remaining case, i.e. c j0 = 1, 2, 3. In this case, C := π(D j ) ⊂ P 2 is a curve in P 2 and its class is [C] = O P 2 (c j0 ). Therefore, by the genus formula on P 2 , we have g a (C) = 1 2 (c j0 -2)(c j0 -1) ≤ 1. Since taking the strict transform of a curve with respect to a birational morphism does not increase its arithmetic genus, we conclude that g a (D j ) ≤ g a (C) ≤ 1.

If g a (D j ) = 1, then c j0 = 3 and m j = 1. In this case, by the genus formula, we have

Proof. Since -K X -[D] = 0 in this case, we have g a (D) = 1 by the genus formula.

Proof. In this case, g a (D j ) = 0 since D is not irreducible. By the genus formula, we have

□

Lemma A.5. D j is orthogonal to v for all j. Moreover, there exists ℓ ∈ Z >0 such that σ ℓ [D j ] = [D j ] for all j.

Proof. Since v is nef and D j is effective, we have

Multiplying both sides by m j and taking the sum for j, we have

However, Since v • (-K X ) = 0, v • D j must be 0 for all j. Therefore, D j ∈ v ⊥ ∩ Pic(X). Since σ preserves the intersection form on the lattice v ⊥ ∩ Pic(X), which is negative definite, there

Lemma A.7. D j is not an exceptional curve of the first kind. In particular, D 2 j ≤ -2. Proof. Since X is minimal, there does not exist an exceptional curve of the first kind C such that C • v = 0 [24, Proposition 4.17]. Since D j is orthogonal to v, D j is not an exceptional curve of the first kind. □ Lemma A.8. For any nontrivial linear combination

Proof. The intersection form is negative definite on the lattice v ⊥ ∩ Pic(X). Therefore, the form is negative definite on the real vector space

For any a j ∈ Z ≥0 , we have

which implies that D is the only effective divisor that belongs to the class -K X . In particular, σ acts on the set {D 1 , . . . , D N } as a permutation.

Proof. By Lemma A.8, the matrix (D i • D j ) i,j is negative definite. Hence, we have h 0 j a j D j = 1 [START_REF] Mase | Studies on spaces of initial conditions for non-autonomous mappings of the place[END_REF]Proposition A.9]. □ Lemma A.10. Let E ⊂ X be an exceptional curve of the first kind. Then, there exists i such that m j D j • E = δ i,j . In particular, D has a component of multiplicity 1.

Proof. By the genus formula, we have E

Then, only one of the following holds:

(1)

and the configuration of D must be one of case (2) in Lemma A. [START_REF] Grammaticos | Discrete Painlevé equations: derivation and properties[END_REF]. Moreover, if rank H 1 (D red ; Z) ≥ 1 and D is not irreducible, then the configuration of D must be either (2) of Lemma A.12 or one of (2) in Lemma A. [START_REF] Grammaticos | Discrete Painlevé equations: derivation and properties[END_REF].

Proof. If m j i = 1 for some i, then, by Lemma A.13, the configuration of D must be one of (2) in Lemma A. [START_REF] Grammaticos | Discrete Painlevé equations: derivation and properties[END_REF]. Let us assume that m j 1 , . . . , m j ℓ ≥ 2 and deduce a contradiction. Take a geometric basis H, E 1 , . . . , E K ∈ Pic(X) and the corresponding blow-down π : X → P 2 . Using this basis, each [D j i ] is expressed as

at most one of c 10 , . . . , c ℓ0 is 1 and the others are 0.

Let us consider the case where

, which is a line in P 2 . In this case, π(D j 2 ), . . . , π(D j ℓ ) are points on H and any intersection among D j 1 , . . . , D j ℓ is transversal. Therefore, when considered as a graph, D j 1 , . . . , D j ℓ ⊂ X form a tree with D j 1 at its root, which contradicts the choice of D j 1 , . . . , D j ℓ since if a tree graph consists of copies of P 1 and every intersection has multiplicity 1, then it does not have a cycle.

If c 10 = • • • = c ℓ0 = 0, then π(D j 1 ), . . . , π(D j ℓ ) are the same point of P 2 . Therefore, D j 1 , . . . , D j ℓ form a tree and any intersection has multiplicity 1, which leads to a contradiction in the same way as above.

In particular, the configuration of D is one of those in Lemma A.12.

Proof. Take a geometric basis H, E 1 , . . . , E K ∈ Pic(X) and the corresponding blow-down π : X → P 2 . Using this basis, [D 1 ] and [D 2 ] are expressed as [START_REF] Arnol | Dynamics of intersections, Analysis, et cetera[END_REF][START_REF] Arnol | d, Dynamics of complexity of intersections[END_REF]. In particular, m 1 must be 1.

Next, we show that

Let us consider the case (c 10 , c 20 ) = (3, 0). Let C = π(D 1 ) ⊂ P 2 and let P = π(D 2 ) ∈ P 2 . Since c 10 = 3, the arithmetic genus of C is 1. Then, C is either a smooth elliptic curve, a rational curve with a node or a rational curve with a cusp. In particular, for any Q ∈ C, the multiplicity of C at Q is at most 2. On the other hand, the multiplicity of C at P is at least Proof. By Lemma A.15, we have

Take a geometric basis H, E 1 , . . . , E K ∈ Pic(X) and the corresponding blow-down π : We may assume that m 1 = 1. Using Lemma A.4, we have

which implies that m 2 = m 3 = 1. Therefore, by Lemma A.13, we have In this case, if m j = 1, then there exists k such that m k = 2 and

Proof. First, we consider the case where D is irreducible. In this case, by Lemma A.3, the arithmetic genus of D is 1. Therefore, D is either a smooth elliptic curve, a rational curve with one nodal singularity, or a rational curve with one cusp. Which case D belongs to is determined by rank H 1 (D red ; Z) ∈ {0, 1, 2}.

From here on, we assume that D is not irreducible. In this case, by Lemma A.2, all the components are smooth rational curves. Therefore, by Lemma A.14, the rank of H 1 (D red ; Z) is 0 or 1.

Let us consider the case where there exist k ̸ = j such that D j • D k ≥ 2. In this case, by Lemma A.15, the configuration of D is one in Lemma A.12. Which case D belongs to is determined by rank H 1 (D red ; Z) ∈ {0, 1}.

If there exist j, k, ℓ such that In what follows we will specify, for three-point mappings from each of the classes I-VI in the classification of QRT-type mappings as it appears in [START_REF] Grammaticos | Full-deautonomisation of a class of second-order mappings in ancillary form[END_REF]Appendix], how to deautonomise a mapping with the singularity confinement property in a way that is sufficiently general for the dynamical degree to be detected by the confinement conditions. Note that the deautonomisations given here are general enough to be sufficient for all possible mappings from the relevant class with the singularity confinement property, but can be specialised further in special cases as we have seen in the examples above. We consider f to be of the form

where k ∈ Z ≥0 and P M and Q N are polynomials of degree M and N respectively with no factors of x. A sufficient deautonomisation is given by taking

where all a 0 , . . . , a M , β 1 , . . . , β N are allowed to evolve with n.