Shrinking targets and recurrent behaviour for forward compositions of inner functions

We prove sharp recurrence results about boundary orbits of forward compositions of inner functions, inspired by fundamental results on iteration of inner functions. Fernández, Melián and Pestana gave a precise version of the Poincaré recurrence theorem for boundary iterates of an inner function that fixes 0. We generalise this to the non-autonomous setting of forward composition sequences Fn = fn ◦ · · · ◦ f1 , n ∈ N, where fn are inner functions that fix 0, establishing an interdependence between the rate of contraction of (Fn) and the size of a shrinking target (In) of arcs in ∂D that (Fn) hits. For a general inner function, Aaronson, and also Doering and Mañé, gave a remarkable dichotomy for boundary iterates. For forward composition sequences, we previously generalised their result in the case that orbits in D approach ∂D sufficiently quickly; here we show that if orbits approach ∂D sufficiently slowly, relative to the rate of contraction, then almost all boundary orbits are dense in ∂D. Finally, we apply our results to wandering domains of transcendental functions, giving new information about the behaviour of iterates on the boundaries of such domains. The techniques we use include Philipp’s version of the second Borel–Cantelli lemma and mixing results of Pommerenke for contracting sequences of inner functions.