Abstract
Given a rational map R, we consider the complement of the postcritical set \(S_R\). In this paper we discuss the existence of invariant Beltrami differentials supported on an R invariant subset X of \(S_R\). Under some geometrical restrictions on X, we show the absence of invariant Beltrami differentials with support intersecting X. In particular, we show that if X has finite hyperbolic area, then X cannot support invariant Beltrami differentials except in the case where R is a Lattes map.
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