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We show that an entire branched cover of finite distortion cannot have acompact branch set if its distortion satisfies a certain asymptotic growthcondition. We furthermore show that this bound is strict by constructing anentire, continuous, open and discrete mapping of finite distortion which ispiecewise smooth, has a branch set homeomorphic to $(n-2)$-dimensional torusand distortion arbitrarily close to the asymptotic bound.

Mappings of finite distortion: compactness of the branch set