PAR–LPTHE 98–22
Algebraic entropy.
M.P. Bellon and CM. Viallet
Laboratoire de Physique Théorique et Hautes Energies
Unité Mixte de Recherche 7589, CNRS et Universités Paris VI et Paris VII.
CERN, Division TH
Abstract:
For any discrete time dynamical system with a rational evolution, we define an
entropy, which is a global index of complexity for the evolution map. We
analyze its basic properties and its relations to the singularities and the
irreversibility of the map. We indicate how it can be exactly calculated.
Postal address: Laboratoire de Physique Théorique et des Hautes Energies
Université Pierre et Marie Curie, boîte postale 126.
4 Place Jussieu/ F–75252 PARIS Cedex 05
1 Introduction
Exploring the behaviour of dynamical systems is an old subject of mechanics [1, 2]. Turning to discrete systems has triggered a huge activity and the notions of sensitivity to initial conditions, numerical (in)stability, Lyapunov exponents and various entropies remained at the core of the subject (see for example [3]).
We describe here the construction of a characterizing number associated to discrete systems having a rational evolution (the state at time is expressible rationally in terms of the state at time ): it is defined in an algebraic way and we call it the algebraic entropy of the map. It is linked to global properties of the evolution map, which usually is not everywhere invertible. It is not attached to any particular domain of initial conditions and reflects its asymptotic behaviour. Its definition moreover does not require the existence of any particular object like an ergodic measure.
In previous works [4, 5, 6, 7], a link has been observed between the dynamical complexity [8] and the degree of the composed map. The naive composition of degree maps is of degree , but common factors can be eliminated without any change to the map on generic points. This lowers the degree of the iterates. For maps admitting invariants, the growth of the degree was observed to be polynomial, while the generic growth is exponential.
We first define the algebraic entropy of a map from the growth of the degrees of its iterates, and give some of its fundamental properties. From the enumeration of the degrees of the first iterates, it is possible to infer the generating function and extract the exact value of the algebraic entropy, even for systems with a large number of degrees of freedom. The reason underlying this calculability is the existence of a finite recurrence relation between the degrees.
After reviewing basic properties of birational maps and of their singularities, we prove such recurrences for specific families. The proof relies on the analysis of the singularities.
2 Algebraic entropy
2.1 Definition
The primary notion we use is the degree of a rational map. In order to assign a well defined degree to a map, we require that all the components of the map are reduced to a common denominator of the smallest possible degree. The maximum degree of the common denominator and the various numerators is called the degree of the rational maps and it is the common degree of the homogeneous polynomials describing the map in projective space. From this definition we obtain the two basic properties:

The degree is invariant by projective transformations of the source and image spaces.

The degree of the composition of two maps is bounded by the product of the degrees of the maps.
From now on, rational maps will always be defined by homogeneous polynomials acting on homogeneous coordinates of the projective completion of affine space. When calculating the composition of two maps, common factors may appear which lower the degree of the resulting map. We then define a reduced composition of and by:
(1) 
We denote by the “true” th iterate of a map , once all factors have been removed.
For a transformation , we can define the sequence of the degrees of the successive iterates of .
Defining proposition
The sequence always admits a limit as . By definition, we call this limit the algebraic entropy of the map .
The proof is straightforward and is a consequence of the inequality .
The algebraic entropy is independent of the particular representation of the rational map . Indeed, if we take the conjugation of by some birational transformation , , the degree of will satisfy for some constant depending on the degree of . A similar inequality can be obtained when writing . In other words, the entropy is a birational invariant associated to .
This quantity can be rather easily computed by taking the images of an arbitrary line. The convergence to the asymptotic behavior is quite fast and can be obtained from the first iterates for which the degree can be exactly calculated.
The growth of measures the complexity of the evolution, since is the number of intersections of the th image of a generic line with a fixed hyperplane. It is related to the complexity introduced by Arnol’d [8], with the difference that we are not dealing with homeomorphisms.
The algebraic entropy also has an analytic interpretation. An invariant Kähler metric exists on the complex projective space and the volume of a dimensional algebraic variety is given by the integral of the th power of the Kähler form. This volume is proportional to the degree of the variety [13]. The area of the image by of a complex line can be expressed as the integral of the squared modulus of the differential of . It is proportional to by the above argument. The algebraic entropy can then be viewed as an averaged exponent: it does not depend on the choice of a starting point and it has the advantage of being of a global nature.
The definition of the algebraic entropy can be generalized to sequences of maps such that the degree of is bounded. We define to be the regularized map . This allows the extension to nonautonomous iterations and to maps which are the product of elementary steps, in which case the sequence is periodic.
In the cases where has a polynomial dependance on , the algebraic entropy is zero, but we can make use of a new invariant, the degree of . As the algebraic entropy, it is a birational invariant.
2.2 Entropy of the Hénon map
For a simple confrontation of the algebraic entropy with more usual approaches, let us consider the much studied Hénon map [14]. Since it is a polynomial map, it is usually considered as having no singularities. This is a misconception: using projective space shows that singularities exist and are located on the line at infinity.
(2)  
(3)  
(4) 
Here is the third homogeneous coordinate. We immediately see on this expression that the line at infinity is sent to the point with homogeneous coordinates . This point is still on the line , so it is a fixed point of the transformation. It will therefore never be mapped to and there cannot be any factorization. The th iterate of this map is of degree and the algebraic entropy is . The remarkable thing is that this number is independent of the parameters and , contrarily to usual dynamical exponents.
3 Birational maps.
Among rational maps, we mainly use birational ones. They are almost everywhere invertible and are therefore quite appropriate for modeling systems possessing a certain amount of reversibility.
3.1 A little bit of algebraic geometry.
Rational relations between two algebraic sets and are relations with a graph which is an algebraic subset of . It would be to restrictive to impose that this define a map from to . In fact, the only algebraic graphs defining maps on the whole space define linear transformations. One therefore only requires that a rational map is one to one on the complement of an algebraic variety, that is a Zariski open set. A birational map defines a bijection from an open subset of to an open subset of .
If we call and the projection on the components of the Cartesian products restricted to , the point will correspond to . When this subset of is not reduced to a point, is by definition in the singular locus of . If we solve for the homogeneous coordinates of the image point, we get homogeneous polynomials in the coordinates of . We therefore get a map defined in . Homogeneity makes it compatible with the scale relation defining projective space. Some vector lines in however are identically mapped to zero: they are projective points without definite images. The set of these points is exactly the singular locus.
If is the homogeneous polynomial representation of a rational map and is a homogeneous polynomial in variables, we denote the pullback of by . It is simply obtained by the composition . The hypersurface of equation is the image by of the hypersurface . If is one of the homogeneous coordinates of , is simply the index component of the polynomial function . Homogeneous polynomials do not define functions on , but sections of a line bundle which only depends on the homogeneity degree.
3.2 Two examples
Let us describe two elementary examples of birational maps. The first one is the generalized Hadamard inverse in . Take two copies of with homogeneous coordinates and and define by the equations in :
(5) 
On the subset where all the are different of zero, we can use affine coordinates by fixing and defines the map
(6) 
If any of the is zero, then all the products must be zero. Let be the set of indices for which is zero. induces a correspondence between and the linear space ( is the hyperplane ). Instead of equations (5), it is often more convenient to give a functional definition of this correspondence. A polynomial definition is:
(7) 
The ’s are polynomial functions of the ’s of degree and they satisfy equations (5). But no formula can give the proper relationship for singular points. For these, at least two of the ’s are zero and therefore all the ’s vanish.
The second example is given in two dimensions by
(8)  
(9) 
with any rational function of . Here again, we can give a homogeneous polynomial formulation. We will have a third variable which will be multiplied by the denominator of .
These two transformations give rise to interesting evolution maps when combined with simple linear transformations. The exchange of the two variables and combined with (8) gives a family of transformation which contains for suitable discrete versions of some Painlevé equations [11]. The transformation (5) and its conjugation by the Fourier transformation yields a birational transformation which appears naturally as a symmetry of the state chiral Potts model [15].
3.3 Singularities
The singular points of a birational map are the vector lines of which are sent to the origin . This singular set is of codimension at least 2. In fact, if there was an algebraic set of codimension 1 sent to the origin, the equation of this set could be factored out of all the components of the image, allowing a reduced description of the map without this singularity.
There is a bigger set where the map is not bijective. Let be a birational map and be its inverse. Than the composition of their representations as polynomial maps in is a map of degree . It is however equivalent to the identity, so that each of the components of the image are of the form , where is a homogeneous polynomial of degree . The set of zeroes of , , is a set where the composition is a priori not defined and it plays a fundamental role.
is an example of a multiplier. When composing two birational maps and , a common factor may appear in the components of . In the case of inverse birational transformations, is the identity and is .
A fundamental property of is that it cannot vanish out of . Otherwise would map an open subset of the set of zeros of to a codimension 1 set where is singular, since is a diffeomorphism outside of . This gives us a contradiction since the singular set of rational maps are of codimension at least 2. Determining the multiplying factor amount to determining the exponents of the different irreducible components of in .
In fact we obtain a definition of the map on a number of apparently singular hypersurfaces, which is a natural continuous extension of the map.
3.4 The meaning of factorization
Consider the successive iterates of a birational map . Suppose we have the following pattern of factorization:
(10)  
(11)  
(12) 
with different from 1. Equation (12) means that the variety is sent to singular points of by . In other words, is blown down to some variety of codimension higher than one by . The latter is non singular for the action of and but is eventually blown up by .
Two situations may occur: it may happen that the image by of the variety is again of codimension 1 and we have a selfregularization of the map. Such a situation was called singularity confinement in [9, 10]. We would rather call it resolution of singularities. Reversibility is recovered on the singular set of after a finite number of time steps. The other possibility is that the image of the variety by remains of codimension larger than one, a situation depicted in figure 1.
Figure 1: A possible blowdown blowup scheme in .
In the scheme of figure 1, the equation of is , and the factor appears anew in . The fifth iterate is regular on .
The drop of the degree of the iterates is due to the presence of singularities on the successive images of a generic surface under the repeated action of . In other words, these images are less and less generic.
4 Recurrence relations for the degree
One of the basic properties of the sequence of degrees is that it seemingly always verifies a finite linear recurrence relation with integer coefficients. If this is true, the algebraic entropy is the logarithm of an algebraic number.
4.1 A simple case in
Consider the map with and given by:
It was used as an example of chaotic behavior in [15] and its singularities have been studied in [4], where the first few elements of the sequence were given:
(13) 
This sequence can be coded in the generating function:
(14) 
The rationality of the generating function is equivalent to the existence of a finite linear recurrence relation for the degrees. The determination of the entropy is straightforward once the recurrence relation is known.
The iterated map is a product of two linearly related involutions and of degree 2. It is useful in this case to look at the sequence of iterates of as the sequence built from .
The possible factorizations come from the fact that the line at infinity () is sent by to a fixed point of and reciprocally. When calculating , will appear as a factor, since send this line to a singular point of .
We want to know the degree of the factor or . The former factor can only contain the factor , but the exponent is not readily known, so we rather examine . We have to determine the curve which is sent by to the line . This is just the first component of the polynomial expression of and can be written . It has degree . In two steps, the line is mapped to a singular point of the following . The curve with equation is therefore mapped to a singular point by and its equation can be factorized in the calculation of . This gives:
(15) 
and consequently the following recurrence relation for :
(16) 
This relation proves formula (14) and yields for the degrees an exponential growth with entropy .
4.2 Factors in the Factors
The previous analysis is simple because the image of remains an irreducible polynomial. This cannot be true in general, since the factors in generally break into pieces under further transformation by [5].
Let us take two birational transformations and with respective inverses and and calculate in two different ways:
(17)  
Since the components of cannot have any common factor, we deduce that divides .
Geometrically, is the equation of a hypersurface which sends to singular points of . Since vanishes on the singular points of , its image vanishes on the zero locus of .
In the example of the previous section, each new factor appearing in is the equation of a hypersurface which sends to the point . The and components of therefore have a common factor . Consequently, the image of by contains the expected factor , while does not contain this factor.
4.3 An example in
Consider the algebra of the finite group , its generic element , with and the generator of .
The algebra has two homomorphic products:
(18)  
(19) 
The product just comes from the product in , and verifies , while . In terms of cyclic matrices, these two products respectively correspond to the matrix product and the element by element (Hadamard) product. The homomorphism between these two products is realized by the Fourier transform.
and will be the two inverses constructed from these products. The components of are the natural coordinates of projective space and and are involutions of degree . and are products of linear factors. These linear factors are the equations of hyperplanes which are sent by the corresponding into points which we call maximally singular.
The important fact is that these maximally singular points are permuted by the other involution. As an example, the maximally singular points of the Hadamard inverse are of the form , i.e., with only one non zero component. The matrix inverse permutes such points by .
If has one vanishing component, say , then will have all its coordinates vanishing except . It follows that is a common factor to all these coordinates. The th coordinate of can be written^{1}^{1}1 As in section 4.1, we write (resp. ) for the composition of alternatively the two inverses (resp. the reduced composition).:
(20) 
The Hadamard inverse is easily calculated on such an expression. The coordinates of are given by:
(21) 
The common factor is simply and this suggest that is a local diffeomorphism on the zeroes of .
We now want to determine the structure of the components of for any . The situation for odd and even will be similar, since the conjugation by the Fourier transform exchanges the two inverses. From the expression (20) of , we see that this point is a generic element of the plane if and only if . We define polynomials generalizing the ’s appearing in (20) such that^{2}^{2}2In this formula, the coordinates are different according to the parity of . They are always such that the following is the Hadamard inverse in those coordinates. :
(22) 
If is the degree of the ’s, then equation (22) yields:
(23) 
The generalization of (21) gives:
(24) 
The factor is therefore of degree . Finally:
(25)  
It is easy from this recurrence relation to determine that for , and therefore are periodic sequences of period 6. The sequence of the ’s is known to have this periodicity. For , is a polynomial of degree 2 in , and for bigger , the sequences are growing like , with the larger root of .
4.4 Another proof
There is another way to prove the previous result, relating directly to the study of the singularities and the blowdown blowup process.
We first need to introduce some notations, using homogeneous coordinates system for . The Hadamard inverse sends into , where . The square of is the multiplication by .
Define to be the projective linear transformation constructed from the matrix
(26) 
with . The inverse of is its complex conjugate . The involution is linearly related to by
(27) 
The product may thus be rewritten , with and . Denote by and the inverses of and respectively.
The maximally singular points of the Hadamard inverse are the points with the only non vanishing component. They are the blow down by of the planes for . They are singular points of and . Denote by the points . The ’s are singular points of and .
We have the following straightforward relations:
(29)  
The relevant singularity structure is entirely described by the two sequences:
(30)  
(31) 
The first squiggly line indicates blow down from hyperplane to point and the last one indicates blow up from point to hyperplane.
Consider now a sequence of varieties of codimension one, constructed by the successive action of , , , and so on. Suppose the ordering is such that acts on the ’s with even index and on the ’s with odd index. The successive images in the sequence are supposed to be regularized by continuity. We denote by the degree of the equation of .
Denote by (resp. ) the order of (resp. ) on . If is the running point of , then we have the defining relations
(32)  
(33) 
Using the fact that and are inverse of each other, and relations (32, 33), we get by evaluating , the following relation on the degrees:
(34) 
Similarly, calculating produces
(35) 
Let and . Relations (34,35) yield
(36) 
From the singularity pattern (30,31), we see that and , so that and . It follows that
(37) 
This combined with (4.4) yields
(38) 
which is the recurrence relation on the degrees of the iterates, with generating function
(39) 
4.5 Discrete Painlevé I
The discrete Painlevé I system is given by the following transformations:
(40) 
where depends on three parameters and is given by . The transformation is just an involution of the form (8) followed by the exchange of and . The homogeneous form is given by:
(41) 
It is easy to obtain that is simply , so that is the only factor which can appear in . The line is sent to the point of coordinates , but it is not sufficient to characterize a possible blowing up. In fact, at leading order in , the image of points approaching this line satisfy the equation . We therefore have to follow the image up to second order. remains a factor in the successive transformations of . In , appears as a factor. This gives a factor of in the transformation of and is the signal according to section 4.2 of the factor appearing in . The factor is however not sufficient to guarantee the factorization of in the next composition. The factorization of depends on the relation which characterizes the form of the given in [11].
We may now establish the recurrence relation obeyed by the degrees. We introduce the polynomials of degree such that the component of is . As in the preceding case, the factor which will replace in the successive factorization is . The factors have disappeared in and the images of are nor singular. Since , we have:
(42) 
whose solution is:
(43)  
(44) 
These results agree with the explicit calculations, producing the sequence of degrees:
(45) 
We can also consider a slight generalization introduced in [6]. The pole part of the transformation of is replaced by a double pole but we do not use variable coefficients.
(46) 
This is now a degree 3 birational map, with . It was shown that we still have the same pattern, but with higher powers of appearing. In , the component gets a factor and we can factorize from . Defining similarly such that the component of is , we get the following recurrence relation for its degree :
(47) 
The solution of this equation allows to recover the results of [6]. The algebraic entropy is given by the logarithm of the largest solution of which is , the square of the golden ratio.
5 Conclusion and perspectives
We have not produced the general proof of the existence of a finite recurrence on the degrees. We have however shown that its origin lies in the singularity structure of the evolution and the possible recovering of reversibility.
In numerous examples which we will not enumerate, we have been able either to establish recurrence equations or to infer a generating function from the first degrees which successfully predict the following ones. This supports the following conjecture.
Conjecture
The generating function of the sequence of the degrees is always a rational function with integer coefficients.
This may even be the case for rational transformation which are not birational [16]. The algebraic entropy is in this case the logarithm of an algebraic number and in the case of vanishing entropy, the sequence of the degrees is of polynomial growth.
There is a keyword which we did not use yet: integrability. Proving integrability in our setting amounts to showing that the motion is a translation on a torus. From the numerous examples we have examined, we believe the algebraic entropy measures a deviation from this type of integrability. We can actually propose the following:
Conjecture 2
If the birational transformation is equivalent to a bijection defined on an algebraic variety deduced from by a finite sequence of blowups, then the sequence of degrees of has at most a polynomial growth.
There also is the question of the relation of the algebraic entropy to other dynamical entropies [3]. The fact that the algebraic maps we study do not necessarily admit an ergodic measure precludes the definition of the KolmogorovSinai entropy in many cases. The most natural correspondance would be with the topological entropy, but requires more work. We must also stress that in any case, the algebraic entropy is a property of the map in the complex domain.
The special properties of rational maps allow to characterize the complexity of the dynamics from the study of a single number, the degree, and to control it through the study of the behaviour of the map in a small number of singular points.
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