Periodic continued fractions and hyperelliptic curves

Periodic continued fractions and hyperelliptic curve

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the $g=2$ case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g\geqslant 1$. The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case ($g=1$), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve $X$, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of $X$. The construction is based on a Lax pair with a spectral curve $S$ of genus four admitting an involution $\sigma$ with two fixed points, and the Jacobian of $X$ arises as the Prym variety Prym $(S,\sigma)$

Laurent Polynomials and Superintegrable Maps

This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations