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A family of integrable maps associated with the Volterra lattice

Hone, Andrew N.W. and Roberts, John A.G. and Vanhaecke, Pol (2023) A family of integrable maps associated with the Volterra lattice. [Preprint] (Submitted) (doi:10.48550/arXiv.2309.02336) (KAR id:103631)


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 ⩾ 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.

Item Type: Preprint
DOI/Identification number: 10.48550/arXiv.2309.02336
Refereed: No
Name of pre-print platform: arXiv
Uncontrolled keywords: Continued fraction, integrable map, Lax pair, Volterra lattice, hyperelliptic curve, Jacobian variety
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations
Q Science > QA Mathematics (inc Computing science) > QA564 Algebraic Geometry
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (
Royal Society (
Depositing User: Andrew Hone
Date Deposited: 01 Nov 2023 17:43 UTC
Last Modified: 03 Nov 2023 15:17 UTC
Resource URI: (The current URI for this page, for reference purposes)

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