Integrable mappings and polynomial growth

TL;DR

This paper studies birational transformations arising from involutions related to solvable models, analyzing their factorization, polynomial growth, and integrability, with some mappings exhibiting elliptic curve behavior and nonlinear recurrences.

Contribution

It classifies six classes of birational involutive transformations, analyzes their factorization properties, and identifies cases with polynomial growth and elliptic curve dynamics, advancing understanding of integrable mappings.

Findings

Some mappings exhibit polynomial growth in complexity.

Certain classes' iterates lie on elliptic curves for q=4.

Homogeneous polynomials satisfy non-trivial nonlinear recurrences.

Abstract

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.