In this paper, we introduce the notion of upper metric mean dimension with potential on any subset (not necessarily compact or invariant) via Carathodory–Pesin structures. We discuss several possible versions of upper measure-theoretic mean dimensions with potential and find conditions to make these notions coincide. In particular, we present a corresponding variational principle and an inverse variational principle.

中文翻译：

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具有子集潜力的上度量平均维度

在本文中，我们通过 Carathodory-Pesin 结构引入了在任何子集（不一定紧凑或不变）上具有潜力的上度量平均维数的概念。我们讨论了具有潜力的上测度理论平均维度的几种可能版本，并找到了使这些概念一致的条件。特别地，我们提出了相应的变分原理和逆变分原理。