  Since its introduction, the method of full deautonomisation by singularity
confinement has proved a strikingly effective way of detecting the dynamical
degrees of birational mappings of the plane. This method is based on an
observed 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 give a proof of this conjectured 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 kind of period map
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.
