The Speiser class \(S\) consists of all entire functions for which the set of critical and asymptotic values is finite. The Eremenko-Lyubich class \(B\) is the larger class of functions for which this set is bounded.
If \(f\in B\) and \(R\) is such that \(\{z:|z|<R\}\) contains all critical and asymptotic values, then \(\Omega =f^{-1}(\{z:|z|>R\})\) is a disjoint union of unbounded simply connected domains \(\Omega_j\) such that \(f:\Omega_j\to \{z:|z| > R\}\) is a covering for all \(j\). The \(\Omega_j\) are called tracts of \(f\). A previous result of the author essentially says that given a union \(\Omega\) of such domains \(\Omega_j\) and a map \(F:\Omega \to \{z:|z|>R\}\) whose restriction to \(\Omega_j\) is a covering for each \(j\), there exists \(f\in B\) such that \(f\) is “close” to \(F\) in some sense. More precisely, \(F=f\circ \varphi\) on \(\Omega\) for some quasiconformal self-map \(\varphi\) of the plane and \(f\circ \varphi\) is bounded in the complement of \(\Omega\). (Some mild assumptions on \(\Omega\) and some additional conclusions on \(\varphi\) and \(f\) are not stated here.)
Here the author studies whether a similar result holds for the Speiser class; that is, whether \(f\) can be chosen to be in \(S\). He shows that one may achieve \(F=f\circ \varphi\) on \(\Omega\) with a quasiconformal map \(\varphi\) for some \(f\in S\). However, for certain choices of \(\Omega\) one may not achieve that \(f\circ \varphi\) is bounded in the complement of \(\Omega\). To achieve that \(f\in S\) it is necessary to introduce additional tracts. It is also shown that \(f\) can be chosen so that the number of tracts of \(f\) is at most twice the number of tracts of \(F\), that is, the number of components of \(\Omega\).