Topological properties of certain iterated entire maps

In this paper it is shown that for a transcendental entire map with certain properties on the distribution of either its zeros, or points in its periodic and preperiodic Fatou domains, all of its Fatou components are simply connected, the union of its Julia set and \(\{ \infty \}\) $\{\infty \}$ is connected and the Julia set is uniformly perfect. Moreover, it is shown that any transcendental entire map that interpolates the 3n+1 problem has only simply connected Fatou components and its Julia set is uniformly perfect.