arXiv subjects

S. Boukraa

Publications and source records attributed to S. Boukraa.

43 records · Page 3Linked to original sources

arXiv2000

Let's Baxterise

We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $ R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the Baxterisation, which amounts to studying the iteration of a birational transformation, is a ``win-win'' strategy: it enables to discard efficiently the non-integrable situations, focusing directly on the two interesting cases where the algebraic varieties are of the so-called ``general type'' (finite order iteration) or are Abelian varieties (infinite order iteration). We emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the Baxter model. We show that the Baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying Yang-Baxter structure is not used: we Baxterise L-operators, local quantum Lax matrices, and quantum Hamiltonians.

hep-th↗

arXiv1999

Real Arnold complexity versus real topological entropy for birational transformations

We consider a family of birational transformations of two variables, depending on one parameter, for which simple rational expressions with integer coefficients, for the exact expression of the dynamical zeta function, have been conjectured. together with an equality between the asymptotic of the Arnold complexity and the exponential of the topological entropy. This identification takes place for the birational mapping seen as a mapping bearing on two complex variables. We revisit this identification between these two quite `universal complexities' by considering now the mapping as a mapping bearing on two real variables. The definitions of the two previous `topological' complexities (Arnold complexity and topological entropy) are modified according to this real-variables point of view. Most of the `universality' is lost, but the identification survives.

chao-dyn↗

arXiv1998

Complexity spectrum of some discrete dynamical systems

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $ 3 \times 3 $ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the $9!$ possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.

chao-dyn↗

arXiv1998

Rational dynamical zeta functions for birational transformations

We propose a conjecture for the exact expression of the dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer coefficients. This yields an algebraic value for the topological entropy. Furthermore the generating function for the Arnold complexity is also conjectured to be a rational expression with integer coefficients with the same singularities as for the dynamical zeta function. This leads, at least in this example, to an equality between the Arnold complexity and the exponential of the topological entropy. We also give a semi-numerical method to effectively compute the Arnold complexity.

chao-dyn↗

arXiv1998

Topological entropy and complexity for discrete dynamical systems

To test a possible relation between the topological entropy and the Arnold complexity, and to provide a non trivial example of a rational dynamical zeta function, we introduce a two-parameter family of two-dimensional discrete rational mappings. The generating functions of the number of fixed points, and of the degree of the successive iterates, are both considered. We conjecture rational expressions with integer coefficients for these two generating functions. and a rational expression for the dynamical zeta function. We then deduce algebraic values for the complexity growth and for the exponential of the topological entropy. Moreover, these two numbers happen to be equal for all the values of the parameters. These conjectures are supported by a semi-numerical method we explain. This method also localizes the integrable cases.

chao-dyn↗

arXiv1994

Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.

hep-th↗

arXiv1993

Towards Three-Dimensional Bethe Ansatz

We introduce a ``pre-Bethe-Ansatz'' system of equations for three dimensional vertex models. We bring to the light various algebraic curves of high genus and discuss some situations where these curves simplify. As a result we describe remarkable subvarieties of the space of parameters.

hep-th↗