Generalized QRT mappings with periodic coefficients

Abstract

We present an extension of the QRT mapping beyond the familiar symmetric and asymmetric varieties. Starting from our results on discrete Painlevé equations, we show that there exist integrable QRT-like mappings, the coefficients of which are periodic functions. We present several examples of mappings with periodic coefficients of various periods and show that there exist cases where the periods are arbitrarily long. We prove the integrability of all the examples by constructing the corresponding conserved quantities and we show how these systems, just as their QRT siblings, can be explicitly integrated in terms of elliptic functions.