Singularities of inner functions associated with hyperbolic maps

Abstract

Let $f$ be a function in the Eremenko-Lyubich class $\mathscr{B}$, and let $U$ be an unbounded, forward invariant Fatou component of $f$. We relate the number of singularities of an inner function associated to $\left.f\right|_{U}$ with the number of tracts of $f$. In particular, we show that if $f$ lies in either of two large classes of functions in $\mathscr{B}$, 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.