Holomorphic Self-maps Research Articles

A hyperbolic-distance inequality for holomorphic maps

We prove an inequality which quantifies the idea that a holomorphic self-map of the disc that perturbs two points is close to the identity function.

Harmonic Measures, Green Potentials and Semigroups of Holomorphic Functions

Let K be a compact subset of the unit disk $\mathbb {D}$. We examine the asymptotic behavior of its trajectory under a semigroup of holomorphic self-maps (ϕt)t≥ 0 of $\mathbb {D}$. More specifically, we obtain results concerning its geometric characteristics such as hyperbolic area, hyperbolic diameter, as well as potential theoretic quantities. Those are the harmonic measure of ϕt(K), its equilibrium measure and its Green equilibrium potential.

A Characterization of Orthogonal Convergence in Simply Connected Domains

Let $${\mathbb {D}}$$ be the unit disc in $${\mathbb {C}}$$ and let $$f:{\mathbb {D}} \rightarrow {\mathbb {C}}$$ be a Riemann map, $$\Delta =f({\mathbb {D}})$$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence $$\{z_n\}\subset \Delta $$ has the property that $$\{f^{-1}(z_n)\}$$ converges orthogonally to a point of $$\partial {\mathbb {D}}$$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $${\mathbb {D}}$$.

On the Koenigs function of semigroups of holomorphic self-maps of the unit disc

Let $$(\phi _t)$$ be a semigroup of holomorphic self-maps of $$\mathbb {D}$$ . In this note, we use an abstract approach to define the Koenigs function of $$(\phi _t)$$ and “holomorphic models” and show how to deduce the existence and properties of the infinitesimal generator of $$(\phi _t)$$ from this construction.

The Product Formula and Evolution Families of Nonexpansive Mappings

In this work we obtain a product formula type for a two-parameters commuting family of nonexpansive mappings on $${\mathbb {D}}$$ . This is established by following the techniques used by Simeon Reich and David Shoikhet in the study of one-parameter semigroups of holomorphic and nonexpansive self-mappings in $${\mathbb {D}}$$ . Also, we stablish such a formula for the family of non-linear resolvent of a strongly $$\rho $$ -monotone functions on $${\mathbb {D}}$$ and its relation with evolution families of nonexpansive mappings on $${\mathbb {D}}$$ . It is worthy mentioning that the product formula is linked with semigroup of linear and nonlinear operators. Also it is associated with the study of vector fields and flows, but in the literature it is established a product formula for time independent flow.

Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc

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 point, the closure of a petal contains either no other boundary fixed points or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of Koenigs functions using divergence rate and universality of models. Moreover, we construct a semigroup having a repelling fixed point in such a way that the intertwining map of the pre-model is not regular.

Loewner–Kufarev Equation for a Strip with an Analogue of Hydrodynamic Normalization

A semigroup of holomorphic self-mappings of a strip with hydrodynamic normalization is studied. A description of infinitesimal transformations of the semigroup is obtained. The differentiability with respect to the time parameter of normalized evolution families is established. An existence theorem is obtained for the evolution equation.

Weighted Differentiation Composition Operators Between Normal Weight Zygmund Spaces and Bloch Spaces in the Unit Ball of $$\hbox {C}^{\mathrm{n}}$$ C n for $$\hbox {n}>1$$ n > 1

Let \(\mu \) be a normal functions on [0, 1), and H(B) be the space of all holomorphic functions on the unit ball B of \(\mathbf C^{n}\). Let \(\varphi \) be a nonconstant holomorphic self-map on B, and \(\psi \) be a holomorphic function on B. The weighted differentiation composition operator \(\psi D_{\varphi }\) is defined on the space H(B) by \(\psi D_{\varphi }(f)=\psi (Rf)\circ \varphi \), for all \(f\in H(B)\). In this paper, the authors characterize the boundedness and compactness of the weighted differentiation composition operator \(\psi D_{\varphi }\) from the normal weight Zygmund space \(Z_{\mu }(B)\) to the normal weight Bloch space \(\beta _{\mu }(B)\) for \(n>1\). As a consequence of the main results, the authors give the briefly sufficient and necessary conditions that the differentiation composition operator \( D_{\varphi }\) is compact from \(Z_{\mu }(B)\) to \(\beta _{\mu }(B)\) for \(\displaystyle {\mu (r)=(1-r)^{s}\log ^{t}\frac{e}{1-r}}\).

A Schwarz lemma at the boundary using the Julia–Wolff–Carathéodory type condition on finite dimensional irreducible bounded symmetric domains

In this paper, we prove a Schwarz lemma at the boundary for holomorphic self-mappings f of finite dimensional irreducible bounded symmetric domains without assuming the boundary regularity of f. Our result generalizes the previous results obtained for holomorphic self-mappings f of the Euclidean unit ball, or of the classical Cartan domains of type I and of type II which are smooth up to the boundary.

Hypercyclic composition operators on the little Bloch space and the Besov spaces

Let S(D) be the collection of all holomorphic self-maps on D of the complex plane C, and Cφ the composition operator induced by φ∈S(D). We obtain that there are no hypercyclic composition operators on the little Bloch space B0 and the Besov space Bp.

Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(\mu)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(\mu)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(\mu)$ and $D_{N,1}(\mu)$ with $N<2n$.

Analytic Semigroups of Holomorphic Mappings and Composition Operators

In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators.

The Schwarz lemma at the boundary of the symmetrized bidisc

The symmetrized bidisc G2 is defined byG2:={(z1+z2,z1z2)∈C2:|z1|<1,|z2|<1,z1,z2∈C}. It is a bounded inhomogeneous pseudoconvex domain without C1 boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carathéodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc G2, and our boundary Schwarz lemma in the paper differs greatly from the earlier related results.

Simultaneous models for commuting holomorphic self-maps of the ball

We prove that a finite family of commuting holomorphic self-maps of the unit ball Bq⊂Cq admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of Bq can commute.

On proper holomorphic self-mappings of generalized complex ellipsoids and generalized Hartogs triangles

In this paper, we study proper holomorphic self-mappings of generalized complex ellipsoids and generalized Hartogs triangles. By making use of our previous result on the holomorphic automorphism group of a generalized complex ellipsoid and Monti-Morbidelli's result on the extendability of a local CR-diffeomorphism between open subsets contained in the strictly pseudoconvex part of the boundary of a generalized complex ellipsoid, we obtain natural generalizations of some results due to Landucci, Chen-Xu and Zapalowski.

Proper holomorphic self-maps of symmetric powers of balls

We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.

HOLOMORPHIC AUTOMORPHISMS AND PROPER HOLOMORPHIC SELF-MAPPINGS OF A TYPE OF GENERALISED MINIMAL BALL

In this paper, we first give a description of the holomorphic automorphism group of a convex domain which is a simple case of the so-called generalised minimal ball. As an application, we show that any proper holomorphic self-mapping on this type of domain is biholomorphic.

Holomorphic mappings of a strip into itself with bounded distortion at infinity

A class of holomorphic self-mappings of a strip which is symmetric with respect to the real axis is studied. It is required that the mappings should boundedly deviate from the identity transformation on the real axis. Distortion theorems for this class of functions are obtained, and domains of univalence are found that arise for certain values of the parameter characterizing the deviation of the mappings from the identity transformation on the real axis.

On the eigenvalues of the infinitesimal generator of a semigroup of composition operators

Suppose that (ϕt) is a one-parameter semigroup of holomorphic self-maps of the unit disk with associated planar domain Ω. Let (Tt) be the corresponding semigroup of composition operators on the classical Hardy space Hp. When the semigroup (ϕt) is hyperbolic, we describe the point spectrum of the infinitesimal generator of (Tt) in terms of the geometry of Ω. The proofs involve various estimates of harmonic measure. We also present various properties of the point spectrum in the parabolic case.

Schwarz Lemma at the Boundary on the Classical Domain of Type $$\mathcal {II}$$ II

Let \(\mathcal {R}_{\mathcal {II}}(n)\) be the classical domain of type \(\mathcal {II}\). We first obtain a sufficient and necessary condition such that the boundary points of \(\mathcal {R}_{\mathcal {II}}(n)\) are smooth. We then establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of \(\mathcal {R}_{\mathcal {II}}(n)\).