Classification of critically fixed anti-Thurston maps

Abstract

We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated to certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurstons's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. Lastly, we give some applications to the global curve attractor and twisting problems, as well as to anti-rational maps with symmetries and to critically fixed anti-polynomials.

Figures (17)

• Figure 1: Constructing a multi-Tischler graph. On the left picture, each of the three gray spheres contains an unobstructed reduced topological Tischler graph, and the blue sphere contains a graph formed by three disjoint edges. The picture in the middle shows the topological space $\mathcal{S}$ obtained by gluing these four spheres along the red edges. Finally, the picture on the right illustrates the $2$-sphere obtained by opening up each seal in $\mathcal{S}$ to a Jordan curve (in red) and the resulting multi-Tischler graph.
• Figure 2: Orientation-reversing setup. The map $f$ is anti-quasiregular, $\widetilde{f}$ is quasiregular, $g$ is anti-rational, and $\widetilde{g}$ is rational. The maps $\phi$, $\psi$, and $\widetilde{\psi}$ are normalized quasiconformal mappings.
• Figure 3: Illustration of the fact that $\sigma_{f^n,Q} = \sigma_{f,Q}^n$. Given a point $\tau_0 \in \mathcal{T}_Q$ represented by a normalized quasiconformal map $\phi_0$, its iterates $\tau_n = \sigma_{f,Q}^n(\tau_0)$ are represented by the unique normalized quasiconformal maps $\phi_n$ for which $g_n = \phi_{n-1} \circ f \circ \phi_n^{-1}$ is anti-rational. This implies that $\phi_n$ is the unique normalized quasiconformal map which makes $G_n = g_1 \circ g_2 \circ \ldots \circ g_n = \phi_0 \circ f^n \circ \phi_n^{-1}$ rational or anti-rational, depending on whether $n$ is even or odd, and thus $\phi_n = \sigma_{f^n,Q}(\phi_0)$.
• Figure 4: Diagram illustrating the argument in Corollary \ref{['cor:symmetric-thurston-maps']} that any symmetry $\theta$ of a marked anti-Thurston map $(f,Q_f)$ is inherited by the equivalent marked anti-rational map $(g,Q_g)$. A normalized combinatorial equivalence between $(f,Q_f)$ and $(g,Q_g)$ yields a normalized combinatorial equivalence between $(f,Q_f)$ and $(g_1, \theta(Q_g))$ with $g_1 = \theta \circ g \circ \theta^{-1}$, so that $g_1 = g$ and $\theta(Q_g) = Q_g$ by uniqueness.
• Figure 5: On the left, a reduced topological Tischler graph $T$ whose associated Schottky map $f_T$ is obstructed. The dashed circle $\gamma$ is a fixed Levy curve, since $f_T^{-1}(\gamma)$ has one connected component $\gamma'$ that is homotopic to $\gamma$ and which is mapped homeomorphically onto $\gamma$. On the right, a dual graph of $T$ is indicated as well, with dashed edges and white vertices. The upper and lower halves of the fixed Levy curve are edges in the dual graph.

Theorems & Definitions (149)

• Theorem A
• Theorem B
• theorem 2.1: Alexander trick
• proof
• remark
• definition 3.1
• remark
• definition 3.2
• remark
• theorem 3.3