Classification of critically fixed anti-rational maps

Abstract

We show that there is a one-to-one correspondence between conjugacy classes of critically fixed anti-rational maps and equivalence classes of certain plane graphs. We furthermore prove that critically fixed anti-rational maps are Thurston equivalent to "topological Schottky maps" associated to these plane graphs, given in each face by a topological reflection in its boundary. As a corollary, we obtain a similar classification of critically fixed anti-polynomials by certain plane trees. One of the main technical tools is an analogue of Thurston's characterization of rational maps in the orientation-reversing case. Contents 1. Introduction 1 2. Review of anti-holomorphic dynamics 2 3. Thurston's theorem for orientation-reversing maps 4 4. Critically fixed anti-rational maps and Tischler graphs 10 5. Plane graphs and Schottky maps 14 6. Critically fixed anti-polynomials 21 7. Examples and Applications 22 References 26

Levy and Thurston obstructions of finite subdivision rules

“…In §9, we define an orientation reversing finite subdivision rule with no edge subdivision from every 2-vertex-connected planar graph G. Then f 2 : (S 2 , Vert(G)) ý and f τ := τ ˝f : (S 2 , Vert(G)) ý are post-critically finite topological branched coverings, where τ is an orientation-reversing automorphism of G. Then we show in Theorem 9.4 that these maps do not have Levy cycles (or equivalently, Thurston obstructions) if and only if G is 3-edge-connected. While this article was being written, two papers [Gey22,LLMM23] were published where it is shown that every critically fixed anti-holomorphic map is constructed in this way and a theorem almost the same as Theorem 9.4 is proved.…”

“…Face-inversion constructions and critically fixed anti-rational maps. The construction in this section was also investigated in [Gey22,LLM22] in the study of critically fixed anti-rational maps.…”

Levy and Thurston obstructions of finite subdivision rules

“…Face-inversion constructions and critically fixed anti-rational maps. The construction in this section was also investigated in [Gey20] and [LLM20] in the study of critically fixed anti-rational maps.…”

“…In Section 9, we define an orientation reversing finite subdivision rule with no edge subdivision from every 2-vertex-connected planar graph G. Then f 2 : pS 2 , VertpGqq ý and f τ :" τ ˝f : pS 2 , VertpGqq ý are Thurston maps, where τ is an orientation-reversing automorphism of G. Then we show in Theorem 9.4 that these maps do not have Levy cycles (or equivalently, Thurston obstructions) if and only if G is 3-edge-connected. While this paper was being written, Geyer published a paper [Gey20] showing that every critically fixed anti-holomorphic map is constructed in this way, where a theorem almost same as Theorem 9.4 is proved.…”

On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps

“…Conversely, given a 2-connected, simple, plane graph Γ with d + 1 vertices, the planar dual of Γ is isomorphic to the Tischler graph of a degree d critically fixed anti-rational map R Γ , which is unique up to Möbius conjugation (cf. [Gey20]). We use H Γ to denote the hyperbolic component containing [R Γ ].…”