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

中文翻译：

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有限失真的映射：分支集的紧致性

我们显示出，如果有限变形的整个变形满足特定渐近生长条件，那么它的整个分支覆盖就不能具有紧凑的分支集。我们进一步表明，通过构造分段变形的完整，连续，开放和离散的有限变形映射，严格限制了边界，该分段变形是平滑的，具有一个分支集，同构为（n -2）维环面，并且变形任意接近渐近边界。