Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that is an n-to-one branched covering with the properties that it extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and it maps each given point on the boundary to the point 1 in the unit circle. We modify a proof by H. Grunsky of Bieberbach's result to show that there is a rational function of 2n+2 complex variables that generates all of these maps. We also show how to generate all the proper holomorphic mappings to the unit disc via the rational function.
In this paper we study commuting families of holomorphic mappings in $\mathbb{C}^n$ which form abelian semigroups with respect to their real parameter. Linearization models for holomorphic mappings are been used in the spirit of Schröder's classical functional equation.
In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.
Let $(ϕ_t)$ be a semigroup of holomorphic self-maps of~$\mathbb D$. In this note, we use an abstract approach to define the Königs function of $(ϕ_t)$ and "holomorphic models" and show how to deduce the existence and properties of the infinitesimal generator of $(ϕ_t)$ from this construction.
We give an explicit relation between the slope of the trajectory of a semigroup of holomorphic functions and the harmonic measure of the associated planar domain ${\varOmega}$. We use this to construct a semigroup whose slope is an arbitrary interval in $ [π/2,-π/2] $. The same method is used for the slope of a backward trajectory approaching a super-repulsive fixed point.
We explore two properties of backward orbits under semigroups of holomorphic self-maps in the unit disk. First, we prove that regular backward orbits are quasi-geodesics for the hyperbolic distance of the unit disk. Then, we show that backward orbits satisfy a useful property, this time in Euclidean terms.
Let $(ϕ_t)_{t \geq 0}$ be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $τ=1$. The angular derivative is $ϕ_t^{\prime}(1)= e^{-λt}$, where $λ\geq 0$ is the spectral value of $(ϕ_t)$. If $λ>0$ the semigroup is hyperbolic, otherwise it is parabolic. Suppose $K$ is a compact non-polar subset of $\mathbb{D}$ with positive logarithmic capacity. We specify the type of the semigroup by examining the asymptotic behavior of $ϕ_t(K)$. We provide a representation of the spectral value of the semigroup with the use of several potential theoretic quantities e.g.…
Let $(ϕ_t)_{t \geq 0}$ be a semigroup of holomorphic functions in the unit disk $\mathbb{D}$ and $K$ a compact subset of $\mathbb{D}$. We investigate the conditions under which the backward orbit of $K$ under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of $K$ are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that $K$ forms with the unit disk.
In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $τ$ of the open unit disk $Δ$. Namely, if $f\in\mathrm{Hol}(Δ,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_{t}\}_{t\geq0}$, we state that the equality $f(z)=o(|z-τ|^{3})$ when $z\toτ$ in each non-tangential approach region at $τ$ implies that $f$ vanishes identically on $Δ$. Note, that if $F$ is a self-mapping of $Δ$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz.…
We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc, the total speed, the orthogonal speed and the tangential speed and show how they are related and what can be inferred from those.
We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.
We show that the tangential speed of a parabolic semigroup of holomorphic self-maps in the unit disc is asymptotically bounded from above by (1/2)logt, proving a conjecture by Bracci. In order to show the proof we need a result of "asymptotical monotonicity" of the tangential speed for proper pairs of parabolic semigroups with positive hyperbolic step.
We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup $(ϕ_t)$ in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to the hyperbolic metric of its Koenigs domain $Ω$ and the latter one deals with Euclidean properties of $Ω$.
Let $(φ_t)$, $(ϕ_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb D\subset \mathbb C$. Let $f:\mathbb D \to \mathbb D$ be a homeomorphism. We prove that, if $f \circ ϕ_t=φ_t \circ f$ for all $t\geq 0$, then $f$ extends to a homeomorphism of $\bar{\mathbb D}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\bar{\mathbb D}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.
It has been an open problem for about ten years whether every trajectory of a parabolic one-parameter semigroup in the unit disk tends to the Denjoy-Wolff point with a definite (and common for all trajectories) slope. In this paper, we give the negative answer to this question.
Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb C^N$. Let $(ϕ_t)$ be a continuous semigroup of holomorphic self-maps of $D$. We prove that if $p\in \partial D$ is an isolated boundary regular fixed point for $ϕ_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point for $ϕ_t$ for all $t\geq 0$. Along the way we also study backward iteration sequences for elliptic holomorphic self-maps of $D$.
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
We study the backward dynamics of one-parameter semigroups of holomorphic self-maps of the unit disk. More specifically, we introduce the speeds of convergence for petals of the semigroup, namely the total, orthogonal, and tangential speeds. These are analogous to speeds of convergence introduced by Bracci, yet profoundly different due to the nature of backward dynamics. Results are extracted on the asymptotic behavior of speeds of petals, depending on the type of the petal. We further discuss the asymptotic behavior of the hyperbolic distance along non-regular backward orbits.
We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc o…
We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy.
In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to ana…
This paper is based on three hours of lectures given by the first author in the "Focus Program on Analytic Function Spaces and their Applications" July 1 -- December 31, 2021, organized by the Fields Institute for Research in Mathematical Sciences. The goal of this paper is to give an introduction to the properties of discrete and continuous $C_0$-semigroups of (weighted) composition operators on various spaces of analytic functions.
We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of $β$-points (i.e. pre-images of values with positive Carleson-Makarov $β$-numbers) of the associated semigroup and of the associated Königs intertwining function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular null poles of the dual generator. Finally we apply such a construction to study ra…
Let $S_{1}=\left\{F_t\right\}_{t\geq 0}$ and $S_{2}=\left\{G_t\right\}_{t\geq 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk $Δ=\{z:|z|<1\}$ generated by $f$ and $g$, respectively. We present conditions on the behavior of $f$ (or $g$) in a neighborhood of a fixed point of $S_{1}$ (or $S_{2}$), under which the commutativity of two elements, say, $F_1$ and $G_1$ of the semigroups implies that the semigroups commute, i.e., $F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all $s,t\geq 0$. As an auxiliary result, we show that the existence of the (angular or unrestricted) $n$…
Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.
Let $(ϕ_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Ω$ be the starlike at infinity domain image of the Koenigs function of $(ϕ_t)$. In this paper we completely characterize the type of convergence of the orbits of $(ϕ_t)$ to the Denjoy-Wolff point in terms of the shape of $Ω$. In particular we prove that the convergence is non-tangential if and only if the domain $Ω$ is `quasi-symmetric with respect to vertical axes'. We also prove that such conditions are equivalent to the curve $[0,…
This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance prop…
We study the backward invariant set of one-parameter semigroups of holomorphic self-maps of the unit disc. Such a set is foliated in maximal invariant curves and its open connected components are petals, which are, in fact, images of Poggi-Corradini's type pre-models. Hyperbolic petals are in one-to-one correspondence with repelling fixed points, while only parabolic semigroups can have parabolic petals. Petals have locally connected boundaries and, except a very particular case, they are indeed Jordan domains. The boundary of a petal contains the Denjoy-Wolff point and, except such a fixed po…
Given a sequence of automorphisms of the polydisk, we show that the associated composition semigroup homomorphisms on the ball of bounded holomorphic functions on the polydisk admit a universal inner function if a certain condition on the automorphisms are satisfied.
If f is a bijection from C^n onto a complex manifold M, which conjugates every holomorphic map in C^n to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of C^n to the semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that ther…
Reinhardt domains determined by their endomorphisms
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University（亚捷隆大学数学与计算机科学学院数学研究所）
We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates the existence and importance of a particular role of the binary operation $\circ$ given by $1 / f \circ g = 1/f + 1/g$ on generators.
This paper is devoted to the study of semigroups of composition operators and semigroups of holomorphic mappings. We establish conditions under which these semigroups can be extended in their parameter to sector given a priori. We show that the size of this sector can be controlled by the image properties of the infinitesimal generator, or, equivalently, by the geometry of the so-called associated planar domain. We also give a complete characterization of all composition operators acting on the Hardy space $H^p$ on the right half-plane.
Let $Δ\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to Δ$ be a Riemann map and let $\{z_k\}\subset Δ$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$ if and only if there exists a simply connected domain $U\subsetneq \mathbb C$ such that $Δ\subset U$ and $Δ$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that i…
In this paper we introduce a class of pseudo-dissipative holomorphic maps which contains, in particular, the class of infinitesimal generators of semigroups of holomorphic maps on the unit ball of a complex Banach space. We give a growth estimate for maps of this class. In particular, it follows that pseudo-dissipative maps on the unit ball of (infinite-dimensional) Banach spaces are bounded on each domain strictly contained inside the ball. We also present some applications.
We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L.Harris via the generation theory for one-parameter semigroups. In particular, we establish geometric and analytic criteria for a holomorphic function on the disk with a boundary null point to be a generator of a semigroup of linear fractional transformations under some relations between three boundary derivatives of the function at this point.
In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish.
Let $(W,Π)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $Π\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ are called {\it $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into $\rS_{1,w_0}({\bar{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an eq…
We present a rigidity property of holomorphic generators on the open unit ball $\mathbb{B}$ of a Hilbert space $H$. Namely, if $f\in\Hol (\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{t\geq 0}$ on $\mathbb{B}$ such that for some boundary point $τ\in \partial\mathbb{B}$, the admissible limit $K$-$\lim\limits_{z\toτ}\frac{f(x)}{\|x-τ\|^{3}}=0$, then $f$ vanishes identically on $\mathbb{B}$.
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, "singular functions on the complex plane" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity wit…
Let D=G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification, and there exists a closed complex subsemigroup, the so-called minimal Olshanskii semigroup, of the complexification characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to it. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Koranyi and Wolf, we give a detailed and complete description of the K-orbit type strata of the minimal Olshanskii semigroup, as K-equivariant fibre bundles. They are given by the conj…
Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show that this semigroup is a quotient of the Teichmueller space of doubly-connected Riemann surfaces by a Z action. Furthermore, the semigroup can be given a complex structure in two distinct, natural ways. We show that these two complex structures are equivalent, and furthermore that multiplication is holomorphic. Finally, we show that the class of quasiconfor…
We establish some estimates of the the angular derivatives from below for holomorphic self-maps of the unit disk at one and two fixed points of the unit circle provided there is no fixed point inside the unit disk. The results complement Cowen-Pommerenke and Anderson-Vasil'ev type estimates in the case of univalent functions. We use the method of extremal length and propose a new semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to receive a new Osserma…
We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.
We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.
We characterize infinitesimal generators of semigroups of holomorphic self-maps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.
We characterize the infinitesimal generator of a semigroup of linear fractional self-maps of the unit ball in $\mathbb C^n$, $n\geq 1$. For the case $n=1$ we also completely describe the associated Koenigs function and we solve the embedding problem from a dynamical point of view, proving, among other things, that a generic semigroup of holomorphic self-maps of the unit disc is a semigroup of linear fractional maps if and only if it contains a linear fractional map for some positive time.
Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {\sl semigroup-fication}, which allows to construct a continuous semigroup $(ϕ_t)$ of holomorphic self-maps of the unit disc whose time one map $ϕ_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $ϕ_1$. Moreover, in case $f$ (and hence $ϕ…
In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup $(\v_t)$ of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark measures techniques. In particular we prove that the Alexandrov-Clark measure of $(\v_t)$ at a boundary regular fixed points is differentiable (in the weak$^\ast$-topology) with respect to $t$.
