Julia sets of inner compositions (Q1924960)

We consider sequences \(\{h_n\}\) of entire functions which converge locally uniformly in the plane to an entire function \(h\), and study the relationship between the sets of normality and nonnormality of the sequence \(\{\Theta_n = h_1 \circ h_2 \circ \cdots \circ h_n\}\) and those of the sequence of iterates of \(h\). In some cases involving polynomials we show that these two sets are the same for both sequences.