On the rate of convergence of parabolic semigroups of holomorphic functions
Let \(\phi_t(z):\mathbb{D}\to \mathbb{D}\), \(t\geq 0\), be a one-parameter continuous semigroup of holomorphic functions. If for some \(t_0\geq 0\), the function \(f_{t_0}\) has no fixed point in \(\mathbb{D}\), then every function of the semigroup has no fixed point in \(\mathbb{D}\) and there exists a unique point \(\tau\in \partial \mathbb{D}\), called the Dejoy-Wolff point of the semigroup \(\phi_t\), such that for every \(z\in \mathbb{D}\), \(\lim_{t\to\infty}\phi_t(z)=\tau\). Without loss of generality, assume \(\tau=1\). The angular derivative \(\phi_t'(1)\) is either \(<1\) for every \(t>0\) or \(=1\) for every \(t>0\). In the former case, the semigroup is called hyperbolic, and in the latter case, it is called parabolic. Moreover, there exists a unique univalent function \(h:\mathbb{D}\to \mathbb{C}\) such that \(h(0)=0\) and \(\phi (t)=h^{-1}(h(z)+t)\). The domain \(\Omega =h(\mathbb{D})\) is called the associated planar domain of the semi-group. The type of the semigroup can be characterized by a geometric property of \(\Omega\). The semigroup is hyperbolic if and only if \(\Omega\) is contained in a horizontal strip. Let \(\nu (\Omega)\) denote the width of the horizontal strip. If the semigroup if parabolic, then \(\Omega\) may be contained in a horizontal half plane of the form \(\{z: \Im z>-\rho\}\) or \(\{z: \Im z<\rho\}\), \(\rho>0\). In a previous paper, the author estimated the rate of convergence if the semigroup is hyperbolic, in terms of the hyperbolic distance. In this paper, he estimates the rate of convergence of the semigroup for both cases in Euclidean distance, as follows. Let \((\phi_t)_{t\geq 0}\) be a semi-group of holomorphic self-maps of \(\mathbb{D}\) with Denjoy-Woff point \(1\) and associated planar domain \(\Omega\). Let \(z\in D\). (a) If the semigroup is hyperbolic, and \(\nu(\Omega)\) is the width of the smallest strip containing \(\Omega\), then there is a constant \(C_1>0\) such that, for every \(t>0\), \[ |\phi_t(z)-1|\leq C_1e^{-\pi t/\nu(\Omega)}. \] (b) If the semigroup is parabolic and \(\Omega\) is contained in a horizontal half plane, then there is a constant \(C_2>0\) such that, for every \(t>0\), \[ |\phi_t(z)-1|\leq \frac{C_2}{t}. \] (c) If the semigroup is parabolic, then there is a constant \(C_3\) such that for every \(t>0\), \[ |\phi_t(z)-1|\leq \frac{C_3}{t^{1/2}}. \]
scientific article; zbMATH DE number 6502336
scientific article