Let \(X\) be a complex manifold. By \(\text{Hol}(X,X)\) we denote the space of holomorphic maps from \(X\) into inself. A one-parameter semigroup of holomorphic maps on \(X\) is a continuous map \(\varphi:\mathbb{R}^ +\to\text{Hol}(X,X)\) such that \(\varphi_ 0=\text{id}_ X\) and \(\varphi_ t\circ\varphi_ t=\varphi_{s+t}\) for all \(s,t\in\mathbb{R}^ +\). Theorem 5. Let \(\varphi:\mathbb{R}^ +\to\text{Hol}(X,X)\) be a one-parameter semigroup on a complex manifold \(X\). Then there is a holomorphic vector field \(F\) on \(X\) such that \[ {\partial\varphi\over\partial t}=F(\varphi). \] In particular, \(\varphi\) is analytic in \(t\). \(F\) is called infinitesimal generator of \(\varphi\). Further let \(H\) be some Finsler metric on \(X\). A holomorphic map \(f\in\text{Hol}(X,X)\) is said to be \(H\)-contraction if \(H(df(\nu))\leq H(\nu)\) for all \(\nu\in TX\). Theorem 8. Let \(H\) be a complete continuous Finsler metric on a complex manifold \(X\). Then a holomorphic vector field \(F\) on \(X\) is the infinitesimal generator of a one-parameter semigroup of \(H\)-contractions iff \(d(H\circ F)\cdot F\leq 0\).