Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

TL;DR

This paper explores boundary dynamics of holomorphic sequences and non-autonomous systems, revealing new behaviors and extending classical results with techniques from geometry, measure, and ergodic theory.

Contribution

It generalizes classical boundary orbit results to sequences of holomorphic maps, uncovering richer dynamics and new phenomena, especially in non-autonomous systems and wandering domains.

Findings

Classical boundary orbit results are extended to holomorphic sequences.

A wider variety of boundary behaviors is demonstrated depending on domain geometry.

New results are obtained even in classical holomorphic dynamics settings.

Abstract

There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the first time, we address such questions in the very general setting of sequences $(F_n)$ of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. Some of our results are new even in the classical setting. Our methods apply in particular to non-autonomous dynamical systems, when $(F_n)$ are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics. The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a `weak independence' version of the second Borel--Cantelli lemma and the concept from ergodic theory of `shrinking targets'.