Birkhoff normalization for a family of superintegrable symplectic maps and its application

Abstract

Birkhoff normalization is considered for a family of analytic symplectic maps near a fixed point. It is proved that, if the family of those maps have an appropriate number of analytic integrals, there exists an analytic system of symplectic (Birkhoff) coordinates in which this family of maps can be solved explicitly. It is applied for a real analytic superintegrable system to show the existence of special coordinates near singular orbits in which the system can be solved explicitly.