Expansion properties for finite subdivision rules II

Floyd, William J.; Parry, Walter; Pilgrim, Kevin M.

doi:10.1090/ecgd/347

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…• A sufficiently large iterate f n of any post-critically finite branched covering f without a Levy cycle is homotopic to a subdivision map [FPP20]. Its 1-skeleton is invariant up to homotopy.…”

Section: Parkmentioning

confidence: 99%

Levy and Thurston obstructions of finite subdivision rules

PARK

2023

Ergod. Th. Dynam. Sys.

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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. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys.18(1) (1998), 221–245] 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.

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Section: Parkmentioning

confidence: 99%

Levy and Thurston obstructions of finite subdivision rules

PARK

2023

Ergod. Th. Dynam. Sys.

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“…‚ Extended Newton graphs for post-critically finite Newton maps [LMS15]. ‚ A sufficiently large iterate f n of any Thurston map f without a Levy cycle [FPP20]. There are Thurston maps which (and whose any iterate) cannot be represented as subdivision maps [FPP20, Section 4].…”

Section: Finite Subdivision Rulesmentioning

confidence: 99%

Levy and Thurston obstructions of finite subdivision rules

Park¹

2020

Preprint

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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.

show abstract