Movable Singularities of Solutions of Difference Equations in Relation to Solvability and a Study of a Superstable Fixed Point

We review applications of exponential asymptotics and analyzable function theory to difference equations in defining an analogue of the Painlevé property for them, and we sketch the conclusions about the solvability properties of first-order autonomous difference equations. If the Painlevé property is present, the equations are explicitly solvable; otherwise, under additional 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 we find that the only cases with the Painlevé property are $a=-2,0,2$ and $4$, for which explicit solutions indeed exist; otherwise, an associated conjugation map develops singularity barriers.