On the rate of convergence of semigroups of holomorphic functions at the Denjoy-Wolff point
For a semigroup \((\phi_t)\) of holomorphic self-maps of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) with Denjoy-Wolff point \(\tau\), the authors study the behavior of \(|\phi_t(z)-\tau|\) as \(t\to\infty\). For \(\phi:\mathbb D\to\mathbb D\) which is holomorphic, neither the identity, nor an elliptic automorphism, there exists a unique Denjoy-Wolff point \(\tau\) such that \(\phi(\tau)=\tau\). In the paper, \(|\tau|=1\) and hence \(\phi(\tau)\) is the angular limit of \(\phi\) at \(\tau\). In this case, the number \[\alpha_{\phi}(\tau):=\liminf_{z\to\tau}\frac{1-|\phi(z)|}{1-|z|},\;\;\;|\tau|=1,\] is the boundary dilatation coefficient of \(\phi\). A semigroup \((\phi_t)\), \(t\geq0\), is a continuous homomorphism \(t\mapsto\phi_t\) with respect to composition, endowed with the topology of uniform convergence on compacta, and \(\tau\) is the Denjoy-Wolff point for all \(\phi_t\). There exists \(\lambda\geq0\), the spectral value of \((\phi_t)\), such that \(\alpha_{\phi_t}(\tau)=e^{-\lambda t}\), \(t\geq0\). A semigroup \((\phi_t)\) is hyperbolic if \(\alpha_{\phi_t}\neq0\), and it is parabolic if \(\alpha_{\phi_t}=0\). Let \(\Omega\) be an open set in \(\mathbb C\) and let \(\Phi_t\) be a group of automorphisms of \(\Omega\). A Koenigs function \(h:\mathbb D\to h(\mathbb D)\subset\Omega\) is univalent, \(h\circ\phi_t=\Phi_t\circ h\) and \(\cup_{t\geq0}\Phi_t^{-1}(h(\mathbb D))=\Omega\). The rate of convergence of \(\phi_t(z)\) to \(\tau\) is given in the following proposition. Proposition 4.1. Let \((\phi_t)\) be a hyperbolic semigroup in \(\mathbb D\) with Denjoy-Wolff point \(\tau\), \(|\tau|=1\), and let \(\lambda\geq0\) be its spectral value. Then \[\lim_{t\to\infty}\frac{1}{t}\log(1-\overline{\tau}\phi_t(z))=-\lambda.\] The main result for hyperbolic semigroups in the paper shows that, in some cases, it is possible to get a tighter rate of convergence to \(\tau\). The authors characterize hyperbolic semigroups for which \[\lim_{t\to\infty}e^{\lambda t}(1-\overline{\tau}\phi_t(z))\in\mathbb C\setminus\{0\},\;\;\;z\in\mathbb D.\] The following theorem is concerned about parabolic semigroups. Theorem 5.3. Let \((\phi_t)\) be a parabolic semigroup in \(\mathbb D\) with Denjoy-Wolff point \(\tau\in\partial\mathbb D\) and let \(h\) be a Koenigs function of the semigroup. Assume that \(h(\mathbb D)\) is contained in a sector \(S_p(\alpha,\beta):=\{p+re^{i\theta}:r>0,-\alpha\pi<\theta<\beta\pi\}\), \(0\leq\alpha,\beta<1\), \(\alpha+\beta>0\), \(p\in\mathbb C\). Then, for every \(z\in\mathbb D\), there exists a constant \(C=C(\alpha,\beta,z)>0\) such that \(|\phi_t(z)-\tau|\leq Ct^{-1/(\alpha+\beta)}\) for all \(t>0\). Conversely, if \(h(\mathbb D)\) contains a sector \(S_p(\alpha,\beta)\) with \(\min\{\alpha,\beta\}>0\), then there exist \(C(z)>0\) and \(T>0\) such that \(|\phi_t(z)-\tau|\geq Ct^{-1/(\alpha+\beta)}\) for all \(t>T\) and \(z\in\mathbb D\).
scientific article; zbMATH DE number 7318497
scientific article