Deautonomisation by singularity confinement and degree growth

In this paper we give an explanation of a number of observations relating to degree growth of birational mappings of the plane and their deautonomisation by singularity confinement. These observations are of a link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity structure to remain unchanged under a sufficiently general deautonomisation. We explain this correspondence for a large class of birational mappings of the plane via the spaces of initial conditions for their deautonomised versions. We show that even for non-integrable mappings in this class, the surfaces forming these spaces have effective anticanonical divisors and one can define a period map parametrising them, similar to that in the theory of rational surfaces associated with discrete Painlev\'e equations. This provides a bridge between the evolution of coefficients in the deautonomised mapping and the induced dynamics on the Picard lattice which encode the dynamical degree.