On the Boundary Behavior of Thurston’s Pullback Map

In the early 1980s, Thurston came up with one of the most important theorems in the field of complex dynamics. Thurston’s characterization theorem gives a criterion for whether a combinatorics (prescribed by a branched cover) can be realized by a rational map, and also provides a rigidity statement: two rational maps are equivalent if and only if they are conjugate by a Mo¨bius transformation.