The stable subset of a univalent self-map

TL;DR

This paper characterizes the stable subset and pre-models of univalent self-maps on Kobayashi hyperbolic cocompact complex manifolds, using the structure of decreasing intersections of complex manifolds.

Contribution

It provides a complete description of the stable subset and pre-models for univalent self-maps on certain complex manifolds, extending understanding of their dynamics.

Findings

Characterization of the stable subset of univalent self-maps.

Description of pre-models in terms of complex structures.

Analysis of decreasing intersections of complex manifolds.

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 $X$.