• 1 Laboratoire de Physique des 2 Infinis Irène Joliot-Curie
• 2 Graduate School of Mathematical Sciences, the University of Tokyo

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

• 37J37 - Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
• 37J70 - Completely integrable discrete dynamical systems
• 39A36 - Integrable difference and lattice equations; integrability tests