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Singularity, complexity, and quasi–integrability of

arXiv:hep-th/9212105v1 17 Dec 1992Singularity, complexity, and quasi–integrability ofrational mappingsG. Falqui∗and C.-M. VialletJuly 1992Laboratoire de Physique Th´eorique et des Hautes EnergiesUnit´e Associ´ee au CNRS (URA 280)Universit´e Paris VI Boˆıte 126Tour 16 – 1er Etage / 4 Place Jussieu75252 PARIS CEDEX 05AbstractWe investigate global properties of the mappings entering the descrip-tion of symmetries of integrable spin and vertex models, by exploitingtheir nature of birational transformations of projective spaces.

We givean algorithmic analysis of the structure of invariants of such mappings.We discuss some characteristic conditions for their (quasi)–integrability,and in particular its links with their singularities (in the 2–plane). Fi-nally, we describe some of their properties qua dynamical systems, mak-ing contact with Arnol’d’s notion of complexity, and exemplify remarkablebehaviours.PAR–LPTHE 92/26To appear in Communications in Mathematical Physics∗Supported in part by Minist`ere de la Recherche et de la Technologie

1IntroductionWe want to analyze in detail some realizations of Coxeter groups [1, 2] bybirational transformations of projective spaces which have been shown to appearin the description of the symmetries of quantum integrable systems [3, 4, 5] [6,7, 8, 9, 10].The first motivation to look at these realizations resides of course in theirrelations with the star–triangle and the Yang–Baxter equations or their higherdimensional generalizations such as the tetrahedron equations. A characteristicfeature of the orbits of the known solutions of the Yang-Baxter equations underthese groups is that they are confined to subvarieties of high codimension of theparameter space (actually curves), signaling the existence of an unexpectedlylarge number of algebraically independent invariants.The discovery and theanalysis of the possible invariants is a decisive step in the study of the Yang–Baxter (tetrahedron,...) equations, in particular for what concerns the so–calledbaxterization problem [11].Another motivation is to use these realizations to construct discrete timeevolution maps, as it is usual in the study a la Poincar´e of dynamical systems,by iterating some element of the group.

One of the main questions in this settingis again to bring to the light the possible presence of invariants and invarianttori [12, 13, 14] [15, 16, 17, 18, 19] [6, 7, 8, 9, 10].If a realization admits algebraic invariants, we will say it has a property ofquasi–integrability.In [6, 7] it was shown, among other things how the direct graphical inves-tigation of the group action by means of numerical calculations (“drawing thepicture”) leads to a nice representation of the orbits. We want to elaborate onthe problem and discuss the deeper structure of the non linear realizations bybirational transformations of projective spaces.We first recall which kind of infinite Coxeter groups arise in the theory ofsolvable models of statistical mechanics and collect some facts about birationalmaps which we shall use in the sequel.We then analyze the cohomological structure of the possible invariants.

Wegive the general (albeit formal) solution to the problem in terms of an algo-rithmic research of linear systems on Pn satisfying a covariance property withrespect to the realizations we handle. We then specialize to the case n = 2,analyze the singular locus of the group, and discuss some illustrative examples.In the final section we make contact with Arnol’d’s notion of complex-ity [20, 21], which measures the growth of the topological non–triviality of theintersection of a fixed subvariety and the image of another one under an iter-ation map.

We will argue how our notion of quasi–integrability is related to apolynomial growth of the complexity while the generic one is exponential.1

2Coxeter groups and birational realizationsOne of the outcomes of [6]-[10] is the construction of a number of groups gen-erated by involutions and of various rational realizations on projective spaces.Consider the Coxeter group G engendered by ν involutions I1, I2, . .

. , Ik, (k =1 .

. .

ν), verifying no relations other than the involution property. The group Gis infinite and there are two essentially different situations.If ν = 2, the group is the infinite dihedral group Z2 ⋉Z, and all elementsmay uniquely be written Iα1 (I1I2)q, with α = 0, 1 and q ∈Z.

The number ofelements of given lenght l is 2.If ν ≥3, the number of elements of lenght l grows exponentially with l, andthe group is in a sense bigger (still countable).As an example for the groups described in [6]-[10], the number ν of generatorsdepends on the dimension d of the lattice: it is just 2d−1 so that if d = 2, G isgenerated by two involutions and if d ≥3, G is generated by more than threeinvolutions.One may then construct various realizations Γ of G by explicit transforma-tions of some projective space. They are obtained by specifying the realizationof the generators.

Since it is precisely the realizations that we want to studyhere, and especially the problem of the existence of invariants, we will mainlytalk about Γ and not G, and use the same notation Ik for the generators of Gand their representatives in Γ.The realizations Γ we consider are essentially obtained from operations onmatrices, especially matrix inversions, and transpositions of their entries, thematrices being originally matrices of Boltzmann weigths of statistical mechani-cal spin and vertex models on the lattice or R–matrices of 2–dimensional fieldtheories. The projective space we consider is just the space of entries of thematrices up to a common multiplicative factor.Let us describe here typical examples of such realizations.Suppose m is a q×q matrix.

The ordinary matrix inverse I defines an involu-tive rational transformation of Pq2−1, which reads in homogeneous coordinates:I :mij −→cofactor of mij(1)We may also consider the element by element inverse (so called Hadamard in-verse):J :mij −→1/mij(2)These two inverses appear in the study of spin models. Notice that I2=J2=1and there is no other relation between I and J.

In particular I and J do notcommute and ϕ = IJ is of infinite order.It is of course possible to define all kinds of block inverses, the size of theblocks ranging from the full matrix size (for I) and 1 (for J).One may also mix I with linear transformations. This happens already whenI and J defined by (1) and (2) are collineated, i.e when there exists a linear2

transformation C such that I = C−1JC (see [6]). We may also mix I withpermutations of the entries.Let us describe here the transpositions of entries which appear in the studyof vertex models on a d–dimensional lattice [9, 10].Suppose M is a multiindex matrix of size qd×qd written in the form M i1i2...idj1j2...jd.There exist d different partial transpositions t1, t2, .

. .

, td with the evident def-inition:(tk M)i1...ik...idj1...jk...jd = M i1...jk...idj1...ik...jd . (3)We clearly have a product and an inverse I for these multiindex matrices.

Wemay define 2(d−1) new inversions by:Iκ = tα1 . .

. tαs I tαs+1 .

. .

tαd = tαs+1 . .

. tαd I tα1 .

. .

tαswhere κ = ({α1, . .

. αs}, {αs+1, .

. .

αd}) is a partition of {1, . .

., d}.These various inverses yield involutive (bi)rational mappings of Pq2d−1. Theyare related by collineations, easy to write from the representation of the partialtranspositions.

Note that the product of all tk’s is the full transposition t andcommutes with all the inverses. Such realizations appear in the study of vertexmodels.We may further enrich the representations by imposing constraints on theentries of the matrices, provided the transformations are compatible with theseconstraints (see [6] for the notion of admissible patterns).

This yields realizationson projective spaces of lower dimensions.Needless to say that along these ideas, one may construct a variety of invo-lutions acting on various projective spaces.We stress that, at the level of the realization, there may exist additionalrelations between the generators, possibly making it finite (as in example (6.4)).3Some facts about rational mappingsIn this section we collect some results about rational and birational mappingsbetween algebraic varieties (see for example [22, 23, 24, 25]).Definition 3.1 A correspondence ϕZ : X →Y between algebraic varieties Xand Y is an algebraic subset Z ⊂X × Y . ϕZ is a rational map if there is aZariski open set U ⊂X on which the correspondence is one to one.If Z ⊂X × Y is a rational map, the inverse correspondence is defined by thegraph Z−1 := {(y, x) ∈Y × X |(x, y) ∈Z}.

If the correspondences Z and Z−1are both rational mappings, then Z (or ϕZ) is called a birational transformation.A birational map is a biholomorphism except on subvarieties of codimension atleast two.3

A linear system D on X is a non empty linear subspace of the space of globalsections of some line bundle L over X. Its base locus is the set of common zeroesof all sections in D. A remarkable result is that [23] there exists a one to onecorrespondence between linear systems on X of dimension d with base locusof codimension not less that 2 and rational maps Xφ−→Pd−1 up to projectiveautomorphisms of Pd−1.For what we are concerned with, the paradigm of rational map is the so–called σ–process or blow up of a point.We refer to [24, 25] for the general definitions.We shall call Hadamardinversion and generically denote by J the prototypical birational mapping inPn.

Let x0, x1, . .

. , xn and y0, y1, .

. .

, yn be coordinates in two different copiesof Pn and let us consider the algebraic set Z ⊂Pn ×Pn given by the n equationsx0y0 = x1y1 = . .

. = xnyn.

By definition, the graph of J is Z.It is valuable to specialize to n = 2: outside the triangle x0x1x2 = 0, Z isthe graph of the map [x0, x1, x2]∼∼∼⊲[1/x0, 1/x1, 1/x2]. The generic point onthe line xi = 0 is sent into the point pi whose only non–vanishing coordinate isyi, while to the points pi corresponds the entire line yi = 0.

One can say thatJ blows up pi to the line xi = 0 and blows down the line xi = 0 to the point pi.Finally, we recall the following properties, which will be used in what follows.From the description of a birational mapping as an algebraic set in the productX × Y it is apparent that ϕ blows up p to a divisor D if and only if ϕ−1blows down D to p, and it is also evident that different points p1 and p2 cannotbe blown up to the same divisor D (otherwise the inverse map would not berational).Also, it is a standard result that blowing up a point adds a free factor Zin the Picard group of X. It follows that if ϕ is a birational map in P2 whichblows up n points, then it must blow down exactly n exceptional divisors, asexplained above.4Invariants and quasi–integrabilityLet Γ be a group of birational transformations in Pn.

A meromorphic function∆: Pn →P1 deserves to be called a Γ–invariant if it satisfies∆(g(x)) = ∆(x)∀g ∈Γ. (4)Since a meromorphic function on an algebraic variety can be thought of as theratio of two sections of a suitable line bundle, we are naturally lead to thefollowing scheme.A Γ one–cocycle is a collection of sections a(g, x) of some line bundles overPn satisfying the cocycle condition:a(g1 g2, x) = a(g1, g2x) · a(g2, x)(5)4

A section σ of a line bundle will be called a–covariant (for some cocycle a) ifthe equationσ(gx) = a(g, x) · σ(x)(6)holds. This equation may be reformulated in group cohomology terms [26] asa = δ σ meaning that a is actually a coboundary.Finding a Γ–invariant is equivalent to finding two sections σ1 and σ2 verifyingequation (6) for the same a.

This means that we are interested in the cobound-aries of the 0–cochains rather than the cohomology groups. The strategy is tofind 1-cocycles admitting a sufficient number of primitives.As a side remark, we notice that these equations, which will play a prominentrole in the sequel, are well defined at all points of Pn.

In fact for any birationaltransformation g, the singular locus is a subvariety of codimension greater thanor equal to two in Pn, and the equations above admit a unique holomorphicextension to the whole of Pn, even if they have meaning in the point set senseonly on the nonsingular locus of g.We are interested in the case where Γ is a Coxeter group, i.e. is generatedby ν involutions Ik of degree dk (the degree is a natural notion in terms ofthe homogeneous coordinates [x0, .

. .

, xn]). A Γ one–cocycle will be completelyspecified by the assignment of ν sections a(Ik, x).Remarkably, the possiblevalues of a(Ik, x) may be found explicitely.Each involution Ik defines a characteristic polynomial φk of degree d2k −1 inthe following manner.

The Ik being involutions, I2k appears as the multiplicationby a degree d2k −1 polynomial φk(x0, . .

. , xn).

We then have the followingLemma 4.1 If a(g, x) is a trivial Γ cocycle, the sections a(Ik, x) divide asuitable power of φi for i = 1, . .

. , ν.Proof.

Suppose a = δ σ. Then, from the coboundary equation (6) for g = I2iwe get that a(I2i , x) = φ(x)m, with m = deg(σ).

The assertion follows from thecocycle condition (5).Definition 4.1 We shall say that Γ is collineated when its generators are allconjugated by means of elements of PGL(n + 1, C) to a standard one, K.This is the case for a number of models among which the Baxter model(n = 3) and the examples of section 6.If the characteristic polynomial φK factorizes into φK = Qjl=0 pdll , then itis clear that for every i there exist a set of global homogeneous coordinates[X(i)0 , . .

. , X(i)n ] in which a(Ii, x) will be the product of the same polynomials pl,possibly weighted with different exponents d′l.Remark.

When K is the Hadamard inversion J in Pn a straightforward com-putation shows that φJ = Qnl=0 x(n−1)lso that we getProposition 4.1 A coboundary for a collineated Coxeter group of birationaltransformations with generators conjugated to the Hadamard inversion J in Pnis a monomial in some suitable homogeneous coordinates, with coefficient ±1.5

It is apparent that our cohomological setting gives an algorithmic prescrip-tion for the search for invariants: find first the characteristic polynomials φk,which is straightforward, then the possible coboundaries, which is a factorizationproblem, and check how many primitives they have, which amounts to solvinga linear problem.The realization Γ in Pn admits p invariants ∆1, . .

. ∆p if there exists ana–covariant linear system Ldp of projective dimension p and degree d for somecocycle a.It is not guaranteed that the orbits of a realization admitting pinvariants lie on subvarieties of dimension n −p.

Indeed the question of thealgebraic independence of the invariants has to be examined further [27].One has to realize how exceptional is the existence of any invariant of thebirational realization, as the following argument shows.Let us consider covariance with respect to one generator K of Γ. SupposeK is of degree dK, and suppose we are looking for an invariant of degree m.Clearly from equation (6) the degree q of the cocycle is related to dK andm by m(dK −1) = q. The dimension of the space P(m, n) of homogeneouspolynomials of degree m in n variables isn+m−1m−1.

The requirement of K–covariance selects a linear system LK in P(m, n), of generic dimension at most12n+m−1m−1. Imposing the same condition for another generator I leads to lookfor the intersection of hyperplanes of at most complementary dimensions inP(m, n).

This intersection is generically empty. As a consequence we haveProposition 4.2 The action of a generic Coxeter group of birational trans-formations in Pn does not admit any non–trivial invariant.In the particular case of collineated realizations, this leads to the further prop-erty, which we prove for ν = 2 for simplicity.Proposition 4.3 The set of collineated groups admitting a non–trivial invari-ant has the structure of a quasi–projective variety.Proof.

The group PGL(n+1, C) admits a natural structure of quasi–projectivevariety, since it is identified with the complement of the degree (n + 1) hyper-surface detA = 0 in P(n+1)2−1. Choosing K as prototypical generator of Γ, I isspecified (in the collineated case) by the choice of an element C ∈PGL(n+1, C)by I = C−1KC.

If for any polynomial P we define PC := P(C−1x) and sety = Cx, then the covariance equation with respect to I readsPC(Ky) = aI(C−1y)PC(y)For each possible cocycle a, this is an algebraic equation in the parameters ta ofthe matrix C. The coordinates of any basis {eα} in LI with respect to a basis inP(m, n) completed from a basis of LK may be expressed as algebraic functions6

of the ta’s. The condition of nontriviality of the intersection is a condition onthe rank of the matrixB :=10eiα(t1, ..t(n+1)2−1).

. .which in turn is an algebraic condition on the eiα(t1, ..t(n+1)2−1), concluding theproof.5Realizations in the 2-planeLet us discuss mappings in P2.

Here our analysis is made quite complete bythe fact that all rational maps from an algebraic surface can be described interms of σ processes and also that singularities of birational maps can occuronly at points. Moreover a theorem of M. Noether [25] assures that the groupof birational maps is generated by the inversion J together with the projectivegroup PGL(3, C).

We will restrict ourselves here to ν = 2, i.e. Γ is generatedby two involutions I and K.Definition 5.1 The singular locus of Γ, S(Γ) is defined asS(Γ) = {x ∈P2|∃g ∈Γ s.t.

g blows x up }Since Γ is generated by I and K, S(Γ) is obtained by the action of Γ on thesingular points of I and K.Definition 5.2 We will say that Γ is properly singular if S(Γ) contains at leastfour points in general position, i.e. such that no three of them are aligned.Let Π(Γ) be the set of singular divisors:Π(Γ) = {π ∈Div(P2)|∃g ∈Γ s.t.

g blows π down to a point }Let us suppose that Γ admits a rational invariant ∆and let us look at whathappens at points in S(Γ). The equation ∆(g(x)) = ∆(x) is clearly meaninglessat the indetermination points of ∆.

Noticing that ∀x ∈S(Γ), ∃πx ∈Π(Γ) suchthat Πx is blown down to x by some element g ∈Γ, we can conclude thatx ∈S(Γ) is either an indetermination point of ∆or ∆is constant along thecorresponding divisor πx, i.e. πx belongs to the pencil of curves ∆= const.Then we can state theProposition 5.1 If S(Γ) is infinite, and Γ is properly singular, then Γ doesnot admit any invariant.7

Proof. With the above observation in mind, the only case we must rule out isthat Γ admits an infinite number of singular points and only a finite numberof singular divisors, since by Bezout’s theorem, any rational degree d invariantadmits at most d2 indetermination points.

Let us suppose that #S(Γ) = ∞and #Π(Γ) = N and let us consider the set P = {Πα} (α = 1 . .

. 2N) of partsof Π(Γ).

We can make a partition of Γ into 2N disjoint subset Γ = SNα=1 Γα,whereΓα = {g ∈Γ|g blows down exactly all the divisors in Πα}At least one of the Γα, say Γ0 is infinite. If we then consider any pair of elements(h0, l0) in Γ0, the product h0 · l−10is a birational map without singularities, andhence an element D ∈PGL(3, C).

Since Γ0 is infinite, we can arrange things sothat D is a non trivial word of even lenght. Since words of odd length in Γ areinvolutions one has:DKD = KDID = I(7)It follows that D permutes the points of S(Γ), and some power Dk must bethe identity.

As a consequence, there is some non–trivial product (IK)l = 1,meaning that actually Γ is finite, and contradicting the infiniteness of S(Γ).Remark. The above proposition proves that a necessary condition for existenceof an invariant is the finiteness of the singular locus.

It is tempting to conjecturethat this is also a sufficient condition for any properly singular realization. Thisis unfortunately not the case, as the example of section (6.6) will show.The next step is to try to relate the singular locus of the group with thesingular locus of the would–be invariant.Let the ∆be a rational invariantfor Γ and let S(∆) be the set of its indetermination points.

If y ∈S(∆) andy /∈S(Γ), the (finite) orbit Γy of y is made out of indetermination points of∆, on which every element g ∈Γ is regular. Since Γy is of even order, say2l, y is a fixed point for (IK)n·l,n ∈Z.

Let ˜Py2pr−→P2 be the 2–plane P2blown up at y. Since we are working with P2, ∆extends to a function ˜∆on ˜Py2with no indetermination points in a neighbourhood of the exceptional divisorE = pr−1(y).

The restriction ϕ = ˜∆|E is a meromorphic map on P1 whosevalue is determined by the limits of ∆(x) along lines in P2 passing through y.We will say that Γ is non–degenerate if for every g ∈Γ the tangent map tog is non–nilpotent at every isolated fixed point of g. Then we can state theProposition 5.2 Let Γ be a non–degenerate Coxeter group of birational trans-formations of P2. If Γ admits an invariant ∆, then S(∆) ⊂S(Γ).Proof.

Suppose y ∈S(∆) and y /∈S(Γ); the non–degeneracy of Γ ensuresthat there is at least one line in PTP 2(y) whose orbit under the tangents to(IK)n·l,l ∈Z is infinite (recall that (IK)l is a minimal length element in Γwhich fixes y). Hence ϕ as defined above assumes the same value on an infinitenumber of points, i.e.

it is constant over the whole of Ey. But this in turn8

implies that ∆is well defined at y which contradicts y ∈S(∆) and ends theproof.Notice that the converse inclusion S(Γ) ⊂S(∆) does not hold in general. Ifthe inclusion is strict, we have seen that for any x ∈S(Γ) and not in S(∆), theinvariant ∆is constant along some divisor πx.

This gives a useful informationabout the invariant (see example 6.3).As for what the singularities of the generic curve Σλ of the pencil are relatedto the singular orbit S(Γ) the following considerations hold. By Bertini’s theo-rem, generic curves are smooth outside the base locus S(∆) whence the chainof inclusionsS(Σλ) ⊂S(∆) ⊂S(Γ)(8)Moreover, exploiting the genus formula for singular curves one can give rela-tions 1 between the degree d of the invariant and S(Γ).

If Σ is a degree d curvein P2 with singular locus Sing(Σ), then [22]g(Σ) = (d −1) · (d −2)2−Xp∈Sing(Σ)δpwith δp depending on the type of the singularity at p. Since Σλ is irreducibleand admits Γ as an infinite group of automorphisms, thanks to the inclusion (8)and to the fact that δp = 0 if p is not in Sing(Σλ), one has(d −1) · (d −2)2−Xp∈S(Γ)δp =n 01This relation shows that there is a balance between the degree of the invariantsand the number and nature of the singular points of the generic curve, andconsequently of Γ.6ExamplesWe describe here some specific examples in P2 with two generators I and J.The physical origin of the models we will be dealing with is to be found notablyamong two–dimensional spin models with interaction along the edges [3], andthis explains the terminology we use.We fix J to be the Hadamard inversion [xi]∼∼∼⊲[1/xi], and I to be I =C−1JC with the collineation matrix C ∈PGL(3, C). We will concentrate onthe parametrization by C and examine some algebraic families in PGL(3, C).For some of them Γ admits a non trivial algebraic invariant.We can associate to Γ a diagram DΓ whose vertices are the points in S(Γ)and where two vertices p1 and p2 ∈SΓ are joined by an edge if either p1 = Jp21We thank M. Talon for useful remarks on this point.9

or p1 = Ip2. The edges are oriented if I (resp.

J) can be applied in one directiononly. The diagram does not characterize the examples, but is a useful tool.

Inparticular the families we give here have been obtained by deformations of givencollineation, demanding the stability of the topology of the diagram.The singular points of J are P1 = [1, 0, 0], P2 = [0, 1, 0], P3 = [0, 0, 1], andthe one of I are Qi = C−1(Pi), (i = 1, 2, 3).One should notice that a number of the families we produce in this wayfall into the general form found in [28] (i.e. verifying C([1, 0, 0]) = [1, 1, 1] andC2([1, 0, 0] = [1, 0, 0]), but with non integer entries.

This general form dependson four parameters:2αβ2−1 + γ−1 −γ2−1 −δ−1 + δ(9)6.1The Z5 familyThe general Z5 (five-state chiral Potts) model is described by a 5 × 5 cyclic ma-trix. We may consider its 2–parameter reduction obtained by imposing that thematrix is symmetric.

It falls into a family of quasi–integrable models parameter-ized by a complex number q, which (whenever this makes sense) is identifiablewith the square root of the number of states [29]. The collineation matrix isCZ5(q) =2q2 −1q2 −12−1 + q−1 −q2−1 −q−1 + q(10)Its singular set is made of ten points:P1, P2, P3 which are the usual singularpoints of J, Q1 = [1, 1, 1] and Q± = [−4, 1 ± q, 1 ∓q], the singular points of I,and R2 = I(P2), R3 = I(P3), R± = J(Q±).

All these points are indeterminationpoints for the invariant∆Zq = (x −z)(y −z)((q −1)(x2 + y2) + 2(q + 1)xy)(2 + (q −2)(zx + zy) + 2xy)(x −y)2Notice that here S(∆) = S(Γ). The diagram DΓ is the following:s✒✑✓✏JQ1s✒✑✓✏IP1s✒✑✓✏JR2s✒✑✓✏JR3s✒✑✓✏R+Js✒✑✓✏R−JsP2I✲sP3I✲sQ+I✲sQ−I✲10

6.2The BMV familyThe matrix of (Boltzmann) weights of the BMV model [6] isxyzyzzzxyzyzyzxzzyyzzxzyzyzyxzzzyzyxThe inversion I is just the matrix inversion. This model pertains to the one–parameter family of quasi–integrable models whose collineation matrix isCBMV (w) =1w −1w1−1010−1(11)and is reached for the value w = 3.

The singular locus is made out of P1, P2, P3andQ1 = [1, 1, 1], Q2 = [w −1, −(w + 1), w −1], Q3 = [1, 1, −1],together with an extra point R = J(Q2) = [w2 −1, (w −1)2, w2 −1]. The casew = 1 is singular.

The family admits the invariant∆BMV (w) =P 2w(x, y, z)Qw(x, y, z)(y + z)4(x −z)2(x −y)wherePw(x, y, z) = (1 −w)(z2 −xy) + (w −3)z(x −y),Qw(x, y, z) = (1 −w2)(y3 −xz2) + (w2 −4w −1)y2(x −z) + 2(w −1)2yz(x −y)Here again S(∆) = S(Γ) and the singular graph is as follows:s✒✑✓✏IP1s✒✑✓✏JQ1sP2s✒✑✓✏JQ2sP3s Q3sR ✒❅❅❅❅✲✲IIJI11

6.3The symmetric Ashkin–Teller modelBy symmetric Ashkin–Teller model we understand the 4–state spin model withthe cyclic and symmetric matrix of Boltzmann weightsx0x1x2x1x1x0x1x2x2x1x0x1x1x2x1x0for which the matrix inversion is collineated to the Hadamard inversion by meansof the matrixCAT =12110−11−21(12)The singular points of the matrix inversion are Q1 = [1, 1, 1], Q2 = [1, 0, −1],Q3 = [1, −1, +1].The singular diagram is drawn belows✒✑✓✏IP1s✒✑✓✏JQ1s✒✑✓✏IP3s✒✑✓✏JQ3sIP2s✎✍☞✌JQ2✲✛It contains a loop connecting the two points P2 and Q2. This means that thepoints P2 = [0, 1, 0] and Q2 = C−1(P2) are not singular for IJ and hence it isno surprise that they are non singular for the invariant∆AT = y2 −xzy(x −z).There is a strict inclusion of S(∆) in S(Γ), since the points P2 and Q2 are notin S(∆).The corresponding divisors ΠP2 = { the line y = 0} (resp ΠQ2 ={ the line x = z}), are indeed curves in the pencil ∆= const.6.4A finite realizationIt is instructive to consider the group described by the collineation matrixCF (q) =1011q−(1 + q)10−1(13)12

which is invertible for q ̸= 0. It is apparent that I and J share P2 = [0, 1, 0] as acommon singular point.

Apart from P2, I admits as singular points Q1 = [1, 1, 1]and Q3 = [q, −(2 + q), −q]. The singular graph DΓ depicted below containstwo more points R = J(Q3) and S = I(Q2).

This model admits at least twoalgebraically independent invariants, which can be taken to be∆(1)F= z4 + x4z2x2∆(2)F= q2zx(y −z)2 + q[(z2 −xy)2 −z(z −2x)(x2 + y2)] + 2(z2 −xy)2(z −y)2(z2 + x2)This, together with Bezout’s theorem tells us that the orbit under Γ of anypoint p ∈P2 is finite and of order not greater that 16, i.e Γ is finite. A directinspection shows that (IJ)4 = 1 and there are additional invariants, of coursenot algebraically independent from the two previous ones.∗P2sSsRsQ3sP3sQ1sP1✒✑✓✏✒✑✓✏JIJ✲II✲J6.5The symmetric Z7 modelThe symmetric Z7 model [6] may be defined by the collineation matrixCZ7 =2662−1 −i√7−1 + i√72−1 + i√7−1 −i√7(14)From the point of view of dynamical systems the model shows chaotic prop-erties in P2.

It has an infinite singular orbit S(Γ). According to proposition5.1, it does not admit any invariant, as it is confirmed by a direct inspection ofthe orbits.

This model behaves actually like a generic element of the family (9).6.6A finite diagram model (FDM)Here Γ is generated by the Hadamard inverse J and I = D−1JD withD =20211−1−11113

Notice that D is not of the form (9). The singular diagram is finite but Γ doesnot have any invariant.

This provides a remarkable example for what we saidin section(5), i.e. that the finiteness of the singular orbit is only a necessarycondition for the existence of a non–trivial invariant.tttttttttt✗✖✔✕✲✛❅❅❅❅❅❅✒❅❅❘✠❅❅■P2Q1P1Q2P3Q3IIJJJJIIIJ7About complexityIn this section we will discus the notion of complexity of the groups we havebeen considering.As we said earlier, we may consider the group G as generated by involutionsand relations.

If the number ν of generators is bigger than 2, G is “exponentiallybig” [30]. We will not comment on this here, and limit ourselves to ν = 2, asone would concentrate on 1–dimensional subgroups of differentiable groups, andexamine the realizations Γ.What we want to point out is that there is a very diverse behaviour of theserealizations , even in the case ν = 2, manifesting itself in the complexity of therepresenting transformations.

The notion of complexity we appeal to is a simpleform of the one introduced in [20].Suppose F is a diffeomorphism of a compact smooth n–manifold M. Let Skand Rl submanifolds of M of dimension respectively k and l. Let Smk be themth iterate of Sk by F and TS,R(m) the intersectionTS,R(m) = Rl ∩Smk .Arnol’d [20] defines the complexity CS,R(m) asCS,R(m) =Xbp(TS,R(m))the bp’s being the Betti numbers. He also proves that for a sufficiently genericchoice of Sk and Rl the complexity grows at most exponentially with m.The analysis of complexity of plane mappings has already been achievedfor the case of polynomial and polynomially invertible transformations [31, 21].We are interested in a wider class of transformations, as the one described insection(5) where Γ is generated by two rational transformations.14

Table 1: Behaviour of degreesModelGrowthrank of invariantsZ5d(k + 1) + d(k −1) −2d(k) periodic of period 3oneBMVd(k + 1) + d(k −1) −2d(k) boundedoneFinited(k) periodic of period 4twoA–Td(k) = 4koneFDMd(k) = 4k (generic)noneZ7d(k + 1) −d(k) = f2k ({fk} a Fibonacci sequence)noneAlthough these transformations are not diffeomorphisms there is a naturalmeasure of the complexity by the degree of iterates.Generically, if the two generating involutions are of degree u and v respec-tively, the degree of ϕ = IK is w = uv, so that deg ϕ(k) = wk. If Σ1 and Σ2are two linear subspaces of P2 then ϕ(k)(Σ1) should be for generic Γ a curve ofdegree wk, and the complexity CΣ1,Σ2(k) would then be exactly wk.However, due to the fact that we are working with projective space, there isa simple mechanism for the lowering of the degree of the iterates ϕ(k), for onehas to factorize out common factors from the expressions of the homogeneouscoordinates of ϕ(k).

This provides a variety of behaviours for the degree d asa function of the order of iteration k, lying between exponential growth andperiodicity, with the particular case of polynomial (or polynomially bounded)growth.The outcome of our analysis is that there is a connection between the ex-istence of an invariant and a polynomial (as opposed to exponential) rate ofgrowth, as shows Table 1, inferred from the results of the direct calculation ofthe first few iterations on the examples of section (6) where the degree of ϕ is4.A more detailed analysis of these properties, together with the study of otherproperties of realizations of Coxeter groups qua dynamical systems such as finite(periodic) orbits is the matter of further investigations (see [32, 33]).Acknowledgments We thank M. Talon, O. Babelon, M. Bellon, J-M. Mail-lard, and G. Rollet for a number of stimulating discussions.References[1] H.S.M. Coxeter and W.O.J.

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