Levy and Thurston obstructions of finite subdivision rules

TL;DR

This paper introduces non-expanding spines for finite subdivision rules to detect Levy cycles and establishes their equivalence with Thurston obstructions under certain conditions, enhancing understanding of polynomial matings.

Contribution

It defines non-expanding spines for subdivision maps and proves the equivalence of Levy and Thurston obstructions for specific polynomial matings.

Findings

Non-expanding spines determine Levy cycles.

Algorithm efficiently detects Levy cycles when subdivision growth is polynomial.

Levy and Thurston obstructions are equivalent for matings involving polynomials with zero core entropy.

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.