Critically fixed Thurston maps: classification, recognition, and twisting

TL;DR

The paper addresses classification, recognition, and twisting for critically fixed Thurston maps on $S^2$ by encoding each map as a blow-up of a planar embedded graph pair $(G,\varphi)$. It proves a canonical bijection between admissible pairs and combinatorial classes of critically fixed Thurston maps, and introduces the charge graph $ ext{Charge}(f)$ together with a lifting algorithm that recovers the model from a map. A global pullback contraction for admissible trees under a given $f$ yields a finite attractor $\,\mathcal{N}_f$, enabling a constructive recognition method and obstruction analysis. The twisting problem is solved in special cases by a graph-rotation operation, linking Dehn twists to combinatorial twists of the model graph and showing periodicity with period dividing $|G\cap \gamma|$. These results unify Thurston theory with graph-based combinatorics, offering algorithmic tools and deepening connections to decomposition theory and Sullivan’s dictionary.

Abstract

An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between Möbius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map $f$ is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair $(G,\varphi)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $\varphi$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $G$. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair $(G,\varphi)$ associated with a critically fixed Thurston map $f$. Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs $(G, \mathrm{id}_{S^2})$.

Figures (18)

• Figure 1: Removing a bigon between two Jordan arcs or curves $\alpha$ and $\beta$.
• Figure 2: Decomposing a Thurston map $f\colon S^2\to S^2$ along a completely invariant multicurve $\Gamma$. The bottom indicates the multicurve $\Gamma = \{\alpha, \beta\}$ and $\mathscr{S}_\Gamma=\{S_1,S_2,S_3\}$. The top illustrates $f^{-1}(\Gamma)$ and $\mathscr{S}_{f^{-1}(\Gamma)}\supset \{S'_1, S'_2, S'_3\}$. The black dots correspond to the postcritical points of $f$. The map $f$ sends each component in $\mathscr{S}_{f^{-1}(\Gamma)}$ onto the component in $\mathscr{S}_\Gamma$ of the same color. At the top, the red curves are isotopic to $\alpha$, the blue curves are isotopic to $\beta$, and the gray curves are non-essential in $(S^2, P(f))$.
• Figure 3: The dynamics of $\widehat{f} \colon \widehat{\mathscr{S}}_\Gamma \to \widehat{\mathscr{S}}_\Gamma$ on small spheres for the example from Figure \ref{['fig: decomposition']}. The black, red, and blue dots correspond to the postcritical points of $f$, the curve $\alpha\in\Gamma$, and the curve $\beta\in\Gamma$, respectively.
• Figure 4: Illustration of the blow-up operation applied to the pair $(G,\varphi)$, where $G\subset S^2$ is the planar embedded graph on the bottom left with four vertices (in black) and two edges (in blue and red), and $\varphi=T_\gamma$ is the Dehn twist about the curve $\gamma$ (in gray). The graph on the bottom right depicts the image $\varphi(G)$. The picture on the top left shows the closed Jordan regions $D_{e_1}\supset e_1$ (in blue) and $D_{e_2}\supset e_2$ (in red), as well as the corresponding open Jordan regions $W_{e_1}\supset D_{e_1}$ and $W_{e_2}\supset D_{e_2}$ with dashed boundaries (in gray). The map $f$ denotes a critically fixed Thurston map on $S^2$ obtained by blowing up the pair $(G,\varphi)$.
• Figure 5: A critically fixed Thurston map $f_\square$ obtained by blowing up the pair $(G_\square, \mathrm{id}_{S^2})$, where $G_\square$ is the planar embedded graph on the left.

Theorems & Definitions (116)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Proposition 1.4
• Corollary 1.5
• Lemma 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5