On the orbits of analytic functions with respect to a Pommiez type operator

Let $\Omega$ be a simply connected domain in the complex plane containing the origin, $A(\Omega)$ be the Fréchet space of all analytic on $\Omega$ functions. An analytic on $\Omega$ function $g_0$ such that $g_0(0)=1$ defines the Pommiez type operator which acts continuously and linearly in $A(\Omega)$. In this article we describe cyclic elements of the Pommiez type operator in space $A(\Omega)$. Similar results were obtained early for functions $g_0$ having no zeroes in domain $\Omega$.