Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in 7]