One may develop the model theory of compact complex manifolds (CCM) with a generic automorphism in rather close analogy to what has been done for existentially closed difference fields, in important work by Chatzidakis and Hrushovski, among others. The corresponding first order theory CCMA is supersimple, and the Zilber trichotomy holds for "finite-dimensional" types of SU-rank 1.

In the talk, I will present some results in CCMA in the spirit of geometric simplicity. I will also discuss how meromorphic dynamical systems may be treated in this framework.