Let \((\phi _t)_{t \ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({{\,\mathrm{{\mathbb {D}}}\,}}\) with Denjoy–Wolff point \(\tau =1\). The angular derivative is \(\phi _t^{\prime }(1)= e^{-\lambda t}\), where \(\lambda \ge 0\) is the spectral value of \((\phi _t)\). If \(\lambda >0\) the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset of \({{\,\mathrm{{\mathbb {D}}}\,}}\). We specify the type of the semigroup by examining the asymptotic behavior of \(\phi _t(K)\). We provide a representation of the spectral value of the semigroup with the use of several potential theoretic quantities, e.g., harmonic measure, Green function, extremal length, condenser capacity.

令\（（\ phi _t）_ {t \ ge 0} \）是单位磁盘\（{{\，\ mathrm {{\ mathbb {D}}} \，}}的全纯自映射的半群\）和Denjoy–Wolff点\（\ tau = 1 \）。角导数为\（\ phi _t ^ {\ prime}（1）= e ^ {-\ lambda t} \），其中\（\ lambda \ ge 0 \）是\（（\ phi _t ）\）。如果\（\ lambda> 0 \），则半群是双曲的，否则它是抛物线的。假设K是\（{{\，\ mathrm {{\ mathbb {D}}} \ ,,}} \）的紧凑非极性子集。我们通过检查\（\ phi _t（K）\）的渐近行为来指定半群的类型。我们使用几个潜在的理论量来表示半群的光谱值，例如谐波测度，格林函数，极值长度，电容器容量。