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.
Invariant graphs of rational maps
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abstract
Let $f$ be a postcritically finite rational map. We prove that, as $n$ large enough, there exists an $f^n$-invariant (finite connected) graph on $\widehat{\mathbb{C}}$ such that it contains the postcritical set of $f$.
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2019 1verdicts
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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.