Every sufficiently large iterate of a Thurston map without Levy cycles and not covered by a torus endomorphism is isotopic to the subdivision map of a finite subdivision rule; the torus-covered case is classified by the eigenvalues of the affine lift.
Pilgrim, Expansion properties for finite subdivision rules I , Sci
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Expansion properties for finite subdivision rules II
Every sufficiently large iterate of a Thurston map without Levy cycles and not covered by a torus endomorphism is isotopic to the subdivision map of a finite subdivision rule; the torus-covered case is classified by the eigenvalues of the affine lift.