Obtaining the growth of higher order mapping through the study of singularities

We examine a collection of integrable and non-integrable birational mappings of order higher than two and show that the methods developed for calculating the degree growth of second order mappings can be transposed to higher-order systems. In particular, Halburd's method which allows one to obtain explicitly the degree growth of a second order mapping, based on the patterns of its singularities, will be shown to work perfectly well in the higher-order case. The only precaution one must take is to choose initial conditions that are specifically adapted to the method. In the case of non-integrable mappings, one can apply either the express variant of Halburd's method or the full-deautonomisation approach and we show that both lead to the (same) exact expression for the dynamical degree of the mapping.