Dynamical approximations of postsingularly finite entire maps

Abstract

We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials $(g_n)$ such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in \mathbb{N}$. To establish this result, we introduce the notion of combinatorial convergence for sequences of entire Thurston maps defined on the topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove that if such a sequence $(f_n)$ converges combinatorially to a Thurston map $f$, then the sequence of Thurston pullback maps $(\sigma_{f_n})$ converges to $\sigma_f$ locally uniformly on the Teichm\"{u}ller space $\mathrm{Teich}(\mathbb{R}^2, A)$.