QRT maps and related Laurent systems
Abstract
In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 12-parameter symmetric QRT map, given by a second-order recurrence, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we first derive systems of two coupled recurrences corresponding to the 5-parameter multiplicative and additive symmetric QRT maps. In all cases, the Laurent property is established using a generalisation of a result due to Hickerson, and exact formulae for degree growth are found from ultradiscrete (tropical) analogues of the recurrences. For the general 18-parameter QRT map it is shown that the components of the iterates can be written as a ratio of quantities that satisfy the same Somos-7 recurrence.
Keywords
Cite
@article{arxiv.1702.07047,
title = {QRT maps and related Laurent systems},
author = {K. Hamad and A. N. W. Hone and P. H. van der Kamp and G. R. W. Quispel},
journal= {arXiv preprint arXiv:1702.07047},
year = {2018}
}
Comments
Typos corrected and extra proposition added with detailed justification of the tropical version in the additive case. Version accepted by Advances in Applied Mathematics