  A unique analytic continuation result is proved for solutions of a relatively
general class of difference equations, using techniques of generalized Borel
summability. This continuation allows for Painlev\'e property methods to be
extended to difference equations. It is shown that the Painlev\'e property (PP)
induces, under relatively general assumptions, a dichotomy within first order
difference equations: all equations with PP can be solved in closed form; on
the contrary, absence of PP implies, under some further assumptions, that the
local conserved quantities are strictly local in the sense that they develop
singularity barriers on the boundary of some compact set. The technique
produces analytic formulas to describe fractal sets originating in polynomial
iterations.
