Levy and Thurston obstructions of finite subdivision rules

Abstract

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

On geometrically finite degenerations II: convergence and divergence

“…Since two geometrically finite polynomials are mateable if and only if the corresponding post-critically finite polynomials are [HT04], the above result can also be proved by directly applying Thurston's characterization theorem of rational maps (see Theorem E in [BD18] and arc intersecting obstruction theorem in [PT98] and [Par21]). It may also be possible to prove similar results using Dylan Thurston's positive criterion [Thu20] (see [DPWY20] for a degree 2 application).…”