The stable subset of a univalent self-map

Abstract

We give a complete description of the stable subset (the union of all backward orbit with bounded step) and of the pre-models of a univalent self-map $f: X\to X$, where $X$ is a Kobayashi hyperbolic cocompact complex manifold, such as the ball or the polydisc in $C^q$. The result is obtained studying the complex structure of a decreasing intersection of complex manifolds, all biholomorphic to $X$.