CNRS, UMR 7644, 91128 Palaiseau
Centre de Physique Theorique, Ecole Polytechnique
Tamizhmani K. M., Grammaticos B., Carstea A. S., Ramani A.
The $q$-discrete Painlevé IV equations and their properties
2004, vol. 9, no. 1, pp. 13-20
We present a detailed study of the properties of two q-discrete Painlevé IV equations: singularity structure, bilinear forms, auto-Bäcklund/Schlesinger transformations, rational solutions as well as special solutions obtained through second- or first-order linear equations.
Lafortune S., Carstea A. S., Ramani A., Grammaticos B., Ohta Y.
Integrable Third-Order Mappings and their Growth Properties
2001, vol. 6, no. 4, pp. 443-448
We study the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painlevé equation to an homographic mapping. We show that the degree grows like $n^3$. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we present a method for the construction of integrable $N$th-order mappings with degree growth $n^N$.