A family of integrable maps associated with the Volterra lattice

TL;DR

This paper explores a family of integrable maps linked to the Volterra lattice, providing explicit formulas, algebraic descriptions, and connections to continued fractions, hyperelliptic curves, and classical integrable systems.

Contribution

It introduces a family of integrable maps associated with hyperelliptic curves, offering explicit formulas, Lax representations, and connections to known integrable systems like the Toda lattice.

Findings

The first map corresponds to genus 2 solutions of the Volterra lattice.

Explicit Hankel determinant formulas for tau functions are derived.

Connections to Somos-5 recurrence and Miura transformations are established.

Abstract

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.