The space of initial conditions for linearisable mappings
  We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties. 
T. Takenawa, M. Eguchi, B. Grammaticos, Y. Ohta, A. Ramani and J.   Satsuma