  Motivated by possible applications to meromorphic dynamics, and generalising
known properties of difference-closed fields, this paper studies the theory
CCMA of compact complex manifolds with a generic automorphism. It is shown that
while CCMA does admit geometric elimination of imaginaries, it cannot eliminate
imaginaries outright: a counterexample to 3-uniqueness in CCM is exhibited.
Finite-dimensional types are investigated and it is shown, following the
approach of Pillay and Ziegler, that the canonical base property holds in CCMA.
As a consequence the Zilber dichotomy is deduced: finite-dimensional types of
SU-rank one are either one-based or almost internal to the fixed field. In
addition, a general criterion for stable embeddedness in TA (when it exists) is
established, and used to determine the full induced structure of CCMA on
projective varieties, simple nonalgebraic complex tori, and simply connected
nonalgebraic strongly minimal manifolds.
