Canonical Models for the Forward and Backward Iteration of Holomorphic Maps

We prove the existence and the essential uniqueness of canonical models for the forward (resp., backward) iteration of a holomorphic self-map f of a cocompact Kobayashi hyperbolic complex manifold, such as the ball \(\mathbb {B}^q\) or the polydisc \(\Delta ^q\). This is done by performing a time-dependent conjugacy of the autonomous dynamical system defined by f, obtaining in this way a non-autonomous dynamical system admitting a relatively compact forward (resp., backward) orbit, and then proving the existence of a natural complex structure on a suitable quotient of the direct limit (resp., subset of the inverse limit). As a corollary we prove the existence of a holomorphic solution with values in the upper half-plane of the Valiron equation for a hyperbolic holomorphic self-map of the unit ball.