Invariant curves of discrete dynamical systems in a neighbourhood of an equilibrium position

The article deals with discrete dynamical systems defined by iterations of a certain smooth map being a local diffeomorphism of a neighbourhood of the coordinate origin, for which the origin is a fixed point. Criteria to the existence of locally invariant curves adherent to the fixed point are obtained and the formulae for the expansion of the above curves into generalised power series are deduced. The author shows also the connection between the existence of those curves and the existence of the so-called asymptotic trajectories, going to the fixed point as the number of iterations infinitely increases or decreases. Of partitucular interest is the problem of the existence of the invariant curves the asymptotic of the motion along which is generalised power one. As an illustration, the author obtainsed criteria to the existence of asymptotic trajectories in several critical cases, when eigen values of the linear approximation matrix lie on a unit circle. The author considers also the problem of the existence of the motions of a material particle in the homogeneous gravity field, which are asymptotic to periodic skips over a critical point of a certain smooth curve.