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There are many classical results, related to the Denjoy--Wolff Theorem,concerning the relationship between orbits of interior points and orbits ofboundary 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 ofsequences $(F_n)$ of holomorphic maps between simply connected domains. We showthat, while some classical results can be generalised, with an interestingdependence on the geometry of the domains, a much richer variety of behavioursis 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 ofwandering domains in holomorphic dynamics.  The proofs use techniques from geometric function theory, measure theory andergodic theory, and the construction of examples involves a `weak independence'version of the second Borel--Cantelli lemma and the concept from ergodic theoryof `shrinking targets'.

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