Limit functions of iterates of entire functions on parts of the Julia set (Q2846920)

Let \(f\) be a non-linear entire function and denote by \(J\) the Julia set of \(f\). For a subset \(E\) of \(J\), let \(\omega(E)\) be the set of all functions from \(E\) to \(J\) which are pointwise limits of a sequence of iterates of \(f\). The corresponding set for uniform limits is denoted by \(\omega_u(E)\).NEWLINENEWLINEThe authors show that for quasi all non-empty compact subsets of \(J\), in the sense of Baire category, the set \(\omega_u(E)\) consists of all continuous functions from \(E\) to \(J\). On the other hand, if \(E\) is a subset of \(J\) of second category, then \(\omega(E)=\emptyset\).