QRT maps and related Laurent systems

TL;DR

This paper explores recursive factorizations of symmetric QRT maps, revealing their Laurent property and degree growth, and connects components of iterates to Somos-7 recurrences.

Contribution

It extends the understanding of QRT maps by deriving coupled recurrences with Laurent property and linking iterates to Somos-7 sequences.

Findings

Recursive factorization yields coupled recurrences with Laurent property.

Exact degree growth formulas are obtained via tropical analogues.

Components of iterates satisfy Somos-7 recurrences.

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.