Preimages under linear combinations of iterates of finite Blaschke products
Abstract
Consider a finite Blaschke product f with f(0) = 0 which is not a rotation and denote by fn its n-th iterate. Given a sequence \an\ of complex numbers, consider the series F(z) = Σn an fn(z). We show that for any w ∈ C, if \an\ tends to zero but Σn |an| = ∞, then the set of points in the unit circle for which the series F 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 tδ, 0 < δ < 1.
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