Parametric representation of univalent functions with boundary regular fixed points (Q1691430)

The author extends a classical result in Loewner's parametric representation of all conformal self-maps from the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) to the semigroup \(\mathfrak U[F]\) of conformal self-maps \(\varphi:\mathbb D\to\mathbb D\) with the prescribed finite set \(F\) of boundary regular fixed points and to \(\mathfrak U_{\tau}[F]\) for \(\varphi\) having the Denjoy-Wolff point at \(\tau\). The class \(\mathfrak U\) of all univalent self-maps of \(\mathbb D\) is a semigroup with respect to composition. A semigroup \((\phi_t)_{t\geq0}\) in \(\mathfrak U\) is defined by the initial value problem \[ \frac{d}{dt}\phi_t(z)=G(\phi_t(z)),\;\;\;t\geq0,\;\;\;\phi_0(z)=z\in\mathbb D, \] where \(G\) is a holomorphic function called the infinitesimal generator of \((\phi_t)\). The set \(T\mathfrak U\) of all infinitesimal generators is a convex cone. For \(\mathfrak S\subset\text{Hol}(\mathbb D,\mathbb D)\), denote by \(T\mathfrak S\) the set of all infinitesimal generators \(G\) giving rise to semigroups \((\phi_t^G)\subset\mathfrak S\). If the angular limit \(\varphi(\sigma):=\angle\lim_{z\to\sigma}\varphi(z)\in\partial\mathbb D\) exists and the angular derivative \(\varphi'(\sigma)\) is finite, the point \(\sigma\) is said to be a regular contact point of the self-map \(\varphi\), and \(\sigma\) is a boundary regular fixed point if, in addition, \(\varphi(\sigma)=\sigma\). For any \(\varphi\in\text{Hol}(\mathbb D,\mathbb D)\setminus\{\text{id}_{\mathbb D}\}\), there exists a unique \(\tau\), \(|\tau|\leq1\), called the Denjoy-Wolff point of \(\varphi\), such that: either \(|\tau|<1\) and \(\varphi(\tau)=\tau\) or \(\tau\) is a boundary regular fixed point of \(\varphi\) with \(\varphi'(\tau)\leq1\). A function \(G:\mathbb D\times[0,\infty)\to\mathbb C\) is called a Herglotz vector field if, for every \(z\in\mathbb D\), the function \([0,\infty)\ni t\mapsto G(z,t)\) is measurable, for a.e. \(t\in[0,\infty)\), \(\mathbb D\ni z\mapsto G(z,t)\in T\mathfrak U\) and, for any compact set \(K\subset\mathbb D\) and any \(T>0\), there exists a nonnegative locally integrable function \(k_{K,T}\) on \([0,\infty)\) such that \(|G(z,t)|\leq k_{K,T}(t)\) for all \(z\in K\) and a.e. \(t\in[0,T]\). It is said that a subsemigroup \(\mathfrak S\subset\text{Hol}(\mathbb D,\mathbb D)\) admits Loewner-type parametric representation if there exists a convex cone \(\mathcal M_{\mathfrak S}\) of Herglotz vector fields in \(\mathbb D\) such that, for every \(G\in \mathcal M_{\mathfrak S}\) and a.e \(t\geq0\), \(G(\cdot,t)\in T\mathfrak S\); for every \(G\in\mathcal M_{\mathfrak S}\), the solution \(\varphi_{s,t}\) to the initial value problem \[ \frac{d}{dt}\varphi_{s,t}(z)=G(\varphi_{s,t}(z),t),\;\;\;\varphi_{s,s}(z)=z, \] satisfies \(\varphi_{s,t}\in\mathfrak S\) for any \(t\geq s\geq0\), and, for every \(\varphi\in\mathfrak S\), there exists \(G\in\mathcal M_{\mathfrak S}\) such that \(\varphi=\varphi_{s,t}\) for some \(t\geq s\geq0\). The main result is given in Theorem 1.1. Theorem 1.1: The following semigroups \(\mathfrak S\) admit Loewner-type parametric representation: \(\mathfrak S:=\mathfrak U[F]\) with \(\text{Card}(F)\leq3\); \(\mathfrak S:=\mathfrak U_{\tau}[F]\) with \(|\tau|=1\) and \(\text{Card}(F)\leq2\); and \(\mathfrak S:=\mathfrak U_{\tau}[F]\) with \(|\tau|<1\) and any finite set \(F\subset\partial\mathbb D\). The problem is also reformulated as a problem of embedding in evolution families, and some results on evolution families \(\mathfrak U[F]\) and \(\mathfrak U_{\tau}[F]\) are presented as well.