Movable singularities of solutions of difference equations in relation to solvability and a study of a superstable fixed point

Abstract

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 Painlevé 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 Painlevé 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 Painlevé 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.