Boundedness of Fatou components of holomorphic maps (Q1768355)

The authors discuss the boundedness of Fatou components of transcendent holomorphic maps composited by any finite holomorphic maps with small growth. Let \(f_1,\dots,f_N\) be nonconstant holomorphic maps in the plane, each having order less than 1/2. They show that if the lower order of \(f_j\) is greater than 0 for some \(j\in \{1,2,\dots,N\}\), then the Fatou set of the map \(h=f_N\circ f_{N-1} \circ \cdots\circ f_1\) has no unbounded components.