Factorization of complex canonical transformations

We investigate the Lie series representation of the canonical transformations in a finite dimensional complex phase space. It is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function, a holomorphic function, and an element of the cyclic group C 4 . The imaginary function can be considered as an observable in the sense of classical mechanics. Some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system. Moreover, a special kind of mappings is studied which provides a link between entropy, action, and state functions. The occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case.