Canonical models for the forward and backward iteration of holomorphic maps

TL;DR

This paper establishes the existence and uniqueness of canonical models for forward and backward iteration of holomorphic self-maps on certain complex manifolds, using conjugacy and limit constructions, with applications to Valiron equations.

Contribution

It introduces a method to construct canonical models for holomorphic self-maps on Kobayashi hyperbolic manifolds, extending the understanding of their dynamical systems.

Findings

Existence of canonical models for forward and backward iteration.

Construction of a natural complex structure on quotient spaces.

Application to holomorphic solutions of Valiron equations.

Abstract

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 performing a time-dependent conjugacy of the dynamical system $(f^n)$, 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 holomorphic self-map of the unit ball.