Suppose \(\left( \phi _t \right) _{t \ge 0}\) is a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) with Denjoy–Wolff point \(\tau =1\). Suppose K is a compact subset of \(\mathbb {D}\). We prove that the capacity of the condenser \((\mathbb {D}, \phi _t(K) )\) is a decreasing function of t. Moreover, we study its asymptotic behavior as \(t \rightarrow + \infty \) in relation with the type of the semigroup.

中文翻译：

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全纯函数和聚光器容量的半群

假设\（\ left（\ phi _t \ right）_ {t \ ge 0} \）是单位磁盘\（\ mathbb {D} \）中具有Denjoy–Wolff点\（\ tau = 1 \）。假设K是\（\ mathbb {D} \）的紧凑子集。我们证明电容器\（（（\ mathbb {D}，\ phi _t（K））\）的容量是t的递减函数。此外，我们研究了与半群类型相关的\（t \ rightarrow + \ infty \）的渐近行为。