{'fraction_non_alphanumeric': 0.03779133564499847, 'fraction_numerical': 0.013518851775446605, 'mean_word_length': 4.442904921165791, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Integrable mappings and the notion of anticonfinement

We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward iterations of a mapping, with only a finite number of iterates taking regular values in between. We show through several concrete examples that the behaviour of some anticonfined singularities is strongly related to the integrability properties of the discrete mappings in which they arise, and we explain how to use this information to decide on the integrability or non-integrability of the mapping.


Introduction
Singularity confinement and algebraic entropy, two well-known integrability detectors, are two of the main factors that can explain the remarkable progress recorded over the past quarter century in the study of discrete integrable systems. Singularity confinement [1] is based on the idea, itself stemming from the analogy to continuous systems, that singularities should play an important role in the integrability of discrete (rational) systems. The confinement requirement consists in demanding that any singularity, the emergence of which depends on the initial conditions, disappear after a finite number of iteration steps rather than propagating indefinitely. The notion of algebraic entropy was introduced by Bellon and Viallet [2]. It is based on ideas of Arnold [3] and Veselov [4], namely that the growth of the complexity of a solution of a given mapping is related to its integrable (or non-integrable) character, integrability being associated with "slow" growth. This argument can be made quantitative when, for rational mappings, one considers the homogeneous degrees of the numerator and denominators of the irreducible fractions that result from this iteration. By iterating the mapping one obtains a sequence of degrees d n for which, as shown in [2], the limit lim n→∞ Both methods just described have their own advantages and drawbacks when used for the detection of integrability. The algebraic entropy approach has the advantage of being easily implemented in practice: in principle one has only to introduce homogeneous coordinates, compute the degree of a sufficient number of iterates of the initial condition in order to obtain a simple "yes" or "no" answer on the integrability of a given mapping. The situation becomes complicated when one considers a multi-parametric system and tries to obtain the integrability constraints on the parameters. Usually, these conditions appear lumped together and are difficult to disentangle. Singularity confinement on the other hand has the advantage of introducing the constraints on the parameters one by one (one for each confinement requirement) leading to a more manageable set of conditions. However, singularity confinement has for years been plagued by its non-sufficient character: as is well-known by now, there exist mappings which are non-integrable despite having confined singularities [5]. Fortunately this "paradox" has recently been resolved [6], thanks to the introduction of what we called the "full-deautonomisation" approach. Standard deautonomisation consists in considering the parameters in a given mapping as functions of the independent variable, and in fixing their form through singularity confinement. The way the latter is implemented is by requiring that the singularity patterns of the autonomous and non-autonomous mappings be the same. (Here we assume that the autonomous mapping that is our starting point indeed has confined singularities). The full-deautonomisation approach is an extension of the standard one where we introduce all possible terms, including terms absent in the initial mapping, the presence of which does not modify the singularity pattern. To put it in a naïve but evocative way: we not only replace 1's by functions of the independent variable, but also selected 0's. Ambiguities in the procedure that may arise from gauge freedom in the definition of the mapping were successfully dealt with in [7].
In [6] we have shown how full-deautonomisation empowers singularity confinement to the point that it becomes a sufficient discrete integrability criterion. What is even more impressive is that this new approach allows one, for all the mappings we examined, to compute their algebraic entropy exactly, a calculation which we confirmed by algebro-geometric analysis. There exist however systems to which full-deautonomisation cannot be applied due to a lack of suitable singularities. In this paper we shall therefore enrich the arsenal of singularity analysis with yet another tool, namely the study of anticonfined singularities. The notion of anticonfinement and its practical usefulness will be explored in the sections that follow.

Enter anticonfinement
The idea of singularity that underlies the singularity confinement approach is that, coming from regular values, at some iteration one loses one degree of freedom. To make this statement more precise, in the case of second-order mappings that we shall focus on in this paper, such a situation arises when the value of the variable x n is such that its iterate x n+1 becomes independent of x n−1 . When the notion of singularity confinement was first proposed, it was also immediately recognised that it would be, de facto, inapplicable to a vast class of mappings in which such singularities cannot appear, in the sense that they cannot arise after a succession of regular values. Polynomial mappings are perhaps the best representatives of this class. Does this mean however that singularity analysis cannot distinguish between a linear (by definition integrable) mapping of the form and the Henon mapping [8] x n+1 + x n−1 = α + βx 2 n , a paradigm of nonintegrability? It turns out that this pessimistic view is unfounded. Let us consider the following initial conditions: x 0 is finite while x 1 is taken to be infinitely large, which, given the form of the mappings, amounts to losing a degree of freedom, namely the value of x 0 . Practically this means that one introduces an infinitesimal quantity ǫ and takes x 1 = ǫ −1 , keeping only the dominant power of ǫ −1 in the subsequent iterations.
Iterating mappings (1) and (2) for x 1 = ǫ −1 (and a general, transcendental, choice of β) we get the following succession of values for (1) for (2). So while for mapping (1) the singularity of x n induced by the initial singular value at x 1 remains of the order of ǫ −1 , for mapping (2) the degree of ǫ −1 grows like 2 |n| . Using this growth we can compute lower bounds for the algebraic entropy of these mappings: the values 0 and log 2 respectively. In fact, in these cases these values turn out to be exactly equal to the algebraic entropy. Singularities (here 'infinities') extending indefinitely both ways from a finite set of regular (here finite) values have been encountered before. The authors of [9] coined the term anticonfinement for such situations.
Confinement indicates the existence of a singularity for some number of iterations, preceded and followed by regular values. Anticonfinement is in a sense the mirror image of this situation, where a few regular values (just one in the examples above) are somehow trapped in the midst of an infinite sequence of singularities. When the notion of anticonfinement was first introduced, it was considered not to be incompatible with integrability. However, as the two examples above show, this reasoning should be refined. The growth of the singularity in an anticonfined situation seems to play an important role, too fast a growth being an indication of nonintegrability.

Anticonfinement and linearisable mappings
The occurrence of anticonfined singularities was first noticed in the context of linearisable mappings. (The authors of [10] used the term "weakly confined" singularities, but we believe that the term "anticonfined" proposed in [9] is more appropriate). Let us illustrate this with an example. The mapping (where a is non-zero and a = 1, lest the mapping become periodic with period 3) is a well-known linearisable system [11]. Its singularity structure is quite simple with two confining patterns {±1, ∓1}. However another singularity does exist. If we start with a finite x 0 and x 1 = ǫ −1 , we find the anticonfining pattern Notice that the singularity always behaves as ǫ −1 , just as in the case of the linear mapping (1). As we mentioned above, the mapping (3) is in fact linearisable. To see this it is convenient to interpret it as a mapping on P 1 × P 1 , after which it is easily seen to be equivalent to the projective mapping (on in the variables u n = (x n + y n )/(x n + 1) and v n = (x n + y n )/(x n − 1).
The fact that a family of linearizable mappings as (3) has confined singularties is not essential for the behaviour of the anticonfined ones. We can illustrate this with the following example. We start from the following triangular mapping on P 1 × P 1 for arbitrary transcendental a (x n+1 , y n+1 ) = x n + y n x n y n + 1 , ay n , which is clearly linearizable. Singularities exist whenever y = ±1 and they are both non-confined. Mapping (4), as it stands, does not have any anticonfined singularities. However, it is straightforward to construct a birationally equivalent mapping that does have such a singularity. Based on the observation that (4) has the fixed point (0, 0), with regular Jacobian matrix J = 1 1 0 a , we introduce the new variable z n = x n /y 2 n . This yields the mapping (y n+1 , z n+1 ) = ay n , z n + y n a 2 y n (z n y 3 n + 1) , which still has non-confined singularities when y = ±1. However, an anticonfined singularity arises when y n = 0 and z n is finite but non-zero: This follows immediately from the linearisation of (4) around its fixed point from which one has that starting from x 0 = ǫ 2 z 0 and y 0 = ǫ one obtains x n = y n = O(ǫ) for all n and hence z n ∼ ǫ −1 except for n = 0. We remark that here again the anticonfined singularity does not grow.
However, another family of linearisable mappings does exist, a very simple example of which is with singularity patterns {±1, 0, ∓1} and an anticonfining pattern where the powers of ǫ −1 increase linearly. This phenomenon is commented upon in [12] in relation to the theory of Diller and Favre [13], which tells us that the existence of such a singularity pattern is tantamount to the existence of an non-confined singularity, in the sense that for such a mapping a finite number of blow-ups (and, if needed, blow-downs) will not give rise to an automorphism on a rational surface (as would have been the case if all the singularities had been confined). These authors also show that, in that case, the mapping has either linear degree growth (which is a characteristic of linearisable mappings), or will have positive algebraic entropy. Hence, in the absence of any other non-confined singularities, we expect linear growth in the anticonfining pattern to imply linear growth for the homogeneous degree for the iterates and thus zero algebraic entropy for the mapping. Such a mapping, moreover, is linearisable.

Solvable mappings
In [14] two of the present authors, in collaboration with C.-M. Viallet, presented a collection of mappings which, while being explicitly solvable, are not integrable since the iteration of a generic initial condition exhibits chaotic behaviour. The principle of construction of these mappings is simple. One starts from a linear equation and uses a transcendental transformation of the dependent variable in order to construct the nonlinear mapping. Let us give a simple example. We start from the linear mapping where k is a positive integer. The solution of (7) is simply w n = αλ n + + βλ n − , where λ ± = (k ± √ k 2 − 4)/2, when k = 2 or w n = αn + β when k = 2. Introducing the transformation x n = exp(w n ) we obtain the reversible rational mapping The case k = 1 will not concern us since its solutions are periodic with period 6, and we concentrate on the cases k = 2 and k > 2. The algebraic entropy of the mapping (8) was calculated in [14] where its value was shown to be log λ + . Thus it vanishes for k = 2, while it is non-zero for k > 2, in agreement with the fact that the k = 2 case is a linearisable mapping while for k > 2 chaotic behaviour is observed numerically. The mapping (8) has only one singularity that can be entered from finite values and in fact, it is an anticonfined one. If we start from x −1 = ǫ and a finite x 0 , iterating forwards and backwards we obtain the following sequence where the powers of ǫ and ǫ −1 increase linearly. For k > 2 on the other hand we find that the anticonfined singularity has rapid growth. For instance, and, in general, we find for the powers of ǫ and ǫ −1 the recursion d n+1 + d n−1 = kd n where d 0 = 0 and d 1 = −1. A lower bound for the algebraic entropy can be obtained from this recursion relation and it turns out that this bound coincides with the exact value of the entropy computed in [14].
We remark that contrary to all the previous examples, the anticonfining pattern for mapping (8) is not symmetrical, and that in general, one might even have (different) finite values on both sides of the pattern. We can illustrate this point, using again mapping (7), by introducing for instance x n = tanh w n which yields a mapping of the type with tanh(2w) = 2x/(1 + x 2 ), tanh(3w) = (x 3 + 3x)/(1 + 3x 2 ) and so on. Here x n+1 is independent of the value of x n−1 whenever x n is equal to ±1, but it turns out that these singularities are also anticonfined. Starting from initial conditions x 0 and x 1 = ±1 + ǫ and iterating forwards and backwards, we obtain at the limit ǫ → 0 the pattern · · · , ∓1, ∓1, ∓1, x 0 , ±1, ±1, ±1, · · · .
Instead of analysing the details of this anticonfining pattern in ǫ, we prefer to introduce the change of variables x n = 1 − y n 1 + y n , so as to send the singularities ±1 of x to 0 and ∞ for y, respectively. Using this transformation for k = 2 and 3 we find precisely the patterns obtained for (8). This is not in the least astonishing since the transform of (9) is nothing but mapping (8).
It is clear that the analysis of any, general, anticonfining pattern can always be simplified by introducing a suitable homographic transformation such that only zeros and/or infinities appear in it.

A non-integrable mapping with confined singularities
In [15] the mapping was studied. It is manifestly not of QRT type [16] and, moreover, it is not integrable, something that can be easily seen by computing the homogeneous degree growth of its iterates. Starting from initial conditions of the form x 0 (of degree 0) and x 1 = p/q and calculating the degrees in (p, q) of the successive iterates, we find the sequence 0, 1,2,4,8,14,24,40,66,108,176,286,464,752,1218, . . .. It is straightforward to verify that for n ≥ 4 these degrees obey the recursion relation d n+1 = 2d n − d n−2 . They therefore grow exponentially with a ratio equal to the golden mean (1 + √ 5)/2, the algebraic entropy being the logarithm of the latter.
When studying the singularity structure of (11) one finds that it has two confined singularities corresponding to the patterns {±1, 0, ∞, ∓1}. As explained in the introduction, the fact that nonintegrability coexists with confined singularities no longer constitutes a paradox thanks to the full-deautonomisation approach. As shown in [6], one must consider the possible non-autonomous extension of (11) where a n is a function of n determined such that the singularity patterns for the autonomous and nonautonomous cases coincide. We thus find for a n the constraint a n+2 = a 2 n a n−1 .
This leads to the characteristic equation: (λ 2 − λ − 1)(λ + 1) = 0 and a growth of log a n with n that is exponential, resulting in lim n→∞ 1 n log log a n = log( 1+ √  5 2 ), i.e. precisely the value of the algebraic entropy. The nonintegrability of this mapping is explained by the existence of yet another singularity, which can be entered from initial conditions where x 1 = 0 (for x 0 free): after 4 iterations of the mapping one reaches (x 4 , x 5 ) = (∞, ∞) which is a fixed point for the mapping (when considered over P 1 × P 1 ). As mentioned before, from [13] it can be understood that this type of singularity is characteristic of mappings for which a finite number of blow-ups (and blow-downs) cannot yield an automorphism on a rational surface. However, this mapping is of a different type than the linearisable one discussed above, its algebraic entropy being positive.
We remark readily that the exponents form a Fibonacci sequence. Hence, here again it is clear that we can use the growth of the anticonfined singularity to obtain a lower bound (equal to the growth rate of the Fibonacci sequence: log( 1+ √  5 2 )) for the algebraic entropy, which in this case coincides with its exact value. From this example and from those in section 4 we can conclude that the exponential growth exhibited by the anticonfined singularity can be used as a non-integrability indicator.

A non-autonomous integrable mapping with an anticonfined singularity
Up to this point we have only studied autonomous mappings. However, nonautonomous mappings might have anticonfined singularities as well. In this section we shall construct a particular integrable mapping in order to show that, even in the nonautonomous case, the existence of an anticonfined singularity with bounded growth does not conflict with the integrability of the mapping.
For this we start from the following nonautonomous mapping y n+1 + y n−1 = 2a n y n y 2 n − 1 (a n = 0), which we shall treat as a mapping on P 1 × P 1 : For this mapping a singularity appears when y n = ±1, which for a generic choice of the function a n will be non-confined. As is well-known however, in case a n satisfies a n+1 − 2a n + a n−1 = 0, these singularities become confined with singularity patterns {±1, ∞, ∓1} and the mapping can be seen to have quadratic degree growth and hence, zero algebraic entropy. In fact, it is a special case of a discrete Painlevé II equation [17].
One could also, for example, choose a n such that it satisfies the first so-called 'late confinement' condition (cf. [18] for a definition) : a n+4 − 2a n+3 + a n+2 − 2a n+1 + a n = 0. In that case the above singularities will still be confined, but the algebraic entropy for the mapping (14) will be equal to the logarithm of the unique root greater than 1 of the polynomial λ 4 − 2λ 3 + λ 2 − 2λ + 1 = 0, i.e. approximately log 1.8832 > 0. The mapping in that case is therefore non-integrable despite having two confined singularities.
The mapping (15) also has singularities when x 2 z = ±1, induced by the rational transformation y n = z n x 2 n from the singularities of (14). For a generic choice of a n these will be non-confined and in general the mapping (15) will be non-integrable. In the case of the late confinement discussed above for (14), the mapping (15) will be nonintegrable despite it having two confined singularities and an anticonfined one with bounded growth, something that can be established in a straightforward way by means of the fulldeautonomisation procedure as explained in [6].
However, when a n satisfies a n+1 − 2a n + a n−1 = 0, the above singularities for x 2 z = ±1 are of course confined and the mapping will have quadratic degree growth. It thus offers an interesting example of a Painlevé type mapping with an anticonfined singularity with bounded growth.

Conclusion
The aim of this paper was to add another tool to the arsenal of singularity analysis, when used as a discrete integrability detector. It goes without saying that our main tool for the investigation of integrability of discrete systems is, as always, the singularity confinement criterion. Thanks to the introduction of the full-deautonomisation approach this criterion has been promoted to a sufficient one. For now, there is no rigorous mathematical proof for this last statement, but there is however ample empirical proof. In fact, using the full-deautonomisation method we have been able to resolve all cases of nonintegrable systems with confined singularities and not a single counterexample to the singularity confinement predictions has been found to date.
While the singularity confinement approach is most powerful, there exist cases where it cannot be applied, for example when the mapping at hand does not possess singularities, in the sense explained above. This is for instance the case for polynomial mappings. In such cases our new approach, based on the study of anticonfined singularities, offers a handy criterion for integrability and complements nicely the confinement approach. As explained in section 2, a singularity is anticonfined if the singular values persist indefinitely in the forwards and backwards iteration of a given system with just a finite region of regular values in the middle. A mapping with an anticonfined singularity is not a priori non-integrable. In order to decide on its integrability, one should study the propagation of the singularity and assess its growth with each iteration. Based on our results we can summarise the different possible situations as follows.
In the case of an anticonfined singularity with zero growth, two cases have to be distinguished. If all other singularities of the mapping are confined then the system at hand may be integrable or not, which can be tested by the full-deautonomisation approach. If on the other hand the mapping has other non-confined singularities, then the system can only be integrable if it is linearisable. As is well-known by now, the confinement property is not necessary in the case of linearisable mappings [19].
The case of an anticonfined singularity with non-zero growth is somewhat simpler. According to [13], if the growth of the anticonfined singularity is linear (and if there are no other non-confined singularities) then we are in the presence of a linearisable mapping. However, if it grows faster (and in fact the only known case corresponds to exponential growth) then the mapping is necessarily non-integrable.
Although in this paper we mainly discussed autonomous mappings, we expect all of the above results to hold as well in the nonautonomous case. Moreover, in all cases presented here (and in many more examples we also studied) it turns out that the logarithm of the growth-rate of the anticonfined singularities coincides with the value of the algebraic entropy (where, in general, it can only offer a lower bound). The general conditions under which this statement can be rigorously shown are an open problem, to which we hope to return in some future work of ours.