Singularities of inner functions associated with hyperbolic maps

Abstract

Let f be a function in the Eremenko-Lyubich class B, and let U be an unbounded, forward invariant Fatou component of f. We relate the number of singularities of an inner function associated to f|U with the number of tracts of f. In particular, we show that if f lies in either of two large classes of functions in B, and also has finitely many tracts, then the number of singularities of an associated inner function is at most equal to the number of tracts of f. Our results imply that for hyperbolic functions of finite order there is an upper bound – related to the order – on the number of singularities of an associated inner function.