Singularities of inner functions associated with hyperbolic maps

TL;DR

This paper investigates the relationship between singularities of inner functions and the structure of hyperbolic maps in the Eremenko-Lyubich class, establishing bounds based on the number of tracts and function order.

Contribution

It establishes a connection between the number of singularities of inner functions and the number of tracts for certain classes of hyperbolic functions in class l, providing bounds related to the function's order.

Findings

Number of singularities is at most the number of tracts for certain classes.

Bound on singularities related to the order of hyperbolic functions.

Results apply to functions with finitely many tracts in class l.

Abstract

Let $f$ be a function in the Eremenko-Lyubich class $\mathcal{B}$, 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 $\mathcal{B}$, 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.