  A unique analytic continuation result is proven for solutions of a relatively
general class of difference equations by using techniques of generalized Borel
summability. We overview applications exponential asymptotics and analyzable
function theory to difference equations, in defining an analog of the
Painlev\'e property for them and we sketch the conclusions with respect to the
solvability properties of first order autonomous ones. It turns out that if the
Painlev\'e property is present the equations are explicitly solvable and in the
contrary case, under further assumptions, the integrals of motion develop
singularity barriers. We apply the method to the logistic map
$x_{n+1}=ax_n(1-x_n)$ where it turns out that the only cases with the
Painlev\'e property are $a=-2, 0, 2$ and 4 for which explicit solutions indeed
exist; in the opposite case an associated conjugation map develops singularity
barriers.
