CRITICALLY FIXED THURSTON MAPS: CLASSIFICATION, RECOGNITION, AND TWISTING

The paper proves a canonical bijection between combinatorial classes of critically fixed Thurston maps and equivalence classes of admissible pairs via the Pilgrim–Tan blow-up and a canonical inverse based on decomposition/charge graphs (Main Theorem A and Theorems 3.29, 3.31). It establishes well-definedness, uniqueness up to equivalence, and naturality under homeomorphisms, including isotopy invariance of the blow-up construction and its compatibility with conjugacies (Propositions 3.2, 3.4, 3.6; Remark 3.7). The candidate solution outlines the same construction and inverse, states the same uniqueness and naturality, and matches the admissibility hypotheses (no isolated vertices; φ(e) isotopic to e rel. vertices). The only nuance is that the statement “Crit equals V(G)” implicitly uses the ‘no isolated vertices’ assumption, which the model includes. Overall, the arguments and structure substantially coincide with the paper’s.