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 $(σ_{f_n})$ converges to $σ_f$ locally uniformly on the Teichmüller space $\mathrm{Teich}(\mathbb{R}^2, A)$.

Figures (9)

• Figure 1: Examples of postsingular portraits
• Figure 2: Admissible quadruple realized by $g_1(z) = \cos(z)$.
• Figure 3: Admissible quadruple realized by $g_2(z) = 2\exp(z^2) - 1$.
• Figure 4: Dynamically admissible quadruple realized by the psf entire map $G_1(z) = \pi \cos (z) / 2$, where $P_{G_1} = \{a_1, a_2, a_3\} = \{-\pi/2, 0, \pi/2\}$.
• Figure 5: Dynamically admissible quadruple realized by the psf entire map $G_2(z) = \sqrt{\ln 2}(1-\exp(z^2))$, where $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.

Theorems & Definitions (111)

• Proposition 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Definition 2.7
• Remark 2.8
• Definition 2.9