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In this paper, we prove that an expanding Thurston map f:S2→S2 is asymptotically h-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is h-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on S2, there exists at least one equilibrium state. For such maps, we also establish...
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