Preimages under linear combinations of iterates of finite Blaschke products

Consider a finite Blaschke product f with \(f(0) = 0\) which is not a rotation and denote by \(f^n\) its n-th iterate. Given a sequence \(\{a_n\}\) of complex numbers, consider the series \(F(z) = \sum _n a_n f^n(z).\) We show that for any \(w \in \mathbb {C},\) if \(\{a_n\}\) tends to zero but \(\sum _n |a_n| = \infty ,\) then the set of points \(\xi \) in the unit circle for which the series \(F(\xi )\) converges to w has Hausdorff dimension 1. Moreover, we prove that this result is optimal in the sense that the conclusion does not hold in general if one considers Hausdorff measures given by any measure function more restrictive than the power functions \(t^\delta ,\) \(0< \delta < 1.\)