Chordal generators and the hydrodynamic normalization for the unit ball (Q901325)

For the Euclidean unit ball \(\mathbb B_n=\{z\in\mathbb C^n:\|z\|<1\}\), a continuous one-real-parameter semigroup of holomorphic functions on \(\mathbb B_n\) is a map \(t\to\Phi_t\in\mathcal H(\mathbb B_n,\mathbb B_n)\), \(t\geq0\), where \(\Phi_0\) is the identity, \(\Phi_{t+s}=\Phi_t\circ\Phi_s\) for all \(t,s\geq0\), and \(\Phi_t\) tends to the identity locally uniformly in \(\mathbb B_n\), when \(t\) tends to 0. Given a semigroup \(\{\Phi_t\}_{t\geq0}\) and a point \(z\in\mathbb B_n\), the limit \[ G(z):=\lim_{t\to0}\frac{\Phi_t(z)-z}{t} \] exists and the vector field \(G:\mathbb B_n\to\mathbb C^n\), called the infinitesimal generator of \(\Phi_t\), is a holomorphic function. The author passes from the unit ball \(\mathbb B_n\) to the Siegel upper half-plane \(\mathbb H_n=\{(z_1,\tilde z)\in\mathbb C^n:\text{Im}(z_1)>\|\tilde z\|^2\}\). Let \(C\) be the Cayley map which maps \(\mathbb H_n\) biholomorphically onto \(\mathbb B_n\). For \(c\geq0\), the class \(\mathcal K(\mathbb H_n,c)\) is the set of all infinitesimal generators \(H\) on \(\mathbb H_n\) satisfying \[ \|H(z)\|_{\mathbb H_n,z}\leq\frac{c}{(-\text{Im}(z_1)+\|\tilde z\|^2)^2}. \] Denote \[ \mathfrak B_n:=\big\{f:\mathbb H_n\to\mathbb H_n\;\big|\;f\;\text{is univalent and}\;f-\text{id}\in\mathcal K(\mathbb H_n,c)\;\text{for some}\;c\geq0\big\}. \] Theorem 4.4. \(\mathfrak B_n\) is a semigroup: if \(f,g\in\mathfrak B_n\), then \(f\circ g\in\mathfrak B_n\). Theorem 4.5. If \(\{H_t\}_{t\geq0}\) is a \(\mathcal K(\mathbb H_n,c)\)-Herglotz vector field, i.e., \(H_t\in\mathcal K(\mathbb H_n,c)\) for almost every \(t\geq0\) and the map \(t\to H_t(z)\) is measurable for every \(z\in\mathbb H_n\), and \(\{\varphi_t\}_{t\geq0}\) is the solution to \[ \frac{\partial\varphi_t(z)}{\partial t}=H_t(\varphi_t(z)),\;\;\varphi_0(z)=z\in\mathbb H_n, \] then \(\varphi_t\in\mathfrak B_n\) for every \(t\geq0\). The author defines smoothness conditions of \(f\in\mathfrak B_n\) near \(\infty\) under which \(\infty\) is the Denjoy-Wolff point of \(f\), \(f\neq\text{id}\). The author also asks several questions, e.g., the following. Question 4.8. Is \(\infty\) the Denjoy-Wolff point for every \(f\in\mathfrak B_n\)?