Symmetries and linearization of ordinary difference equations

The connection between symmetries and linearizations of discrete-time dynamical systems is being inverstigated. It is shown, that existence of semigroup structures related to the vector field and having linear representations enables reduction of linearization problem to a system of first order partial differential equations. By means of inverse of the Poincare map one can relate symmetries in such linearizable systems to continuous and discrete ones of the corresponding differential equations.