Denjoy-Wolff Point Research Articles

Characterization of finite shift via Herglotz's representation

A complete characterization of parabolic self-maps of finite shift is given in terms of their Herglotz's representation. This improves a previous result due to Contreras, Díaz-Madrigal, and Pommerenke. We also derive some consequences for the rate of convergence of these functions to their Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and Zarvalis for the continuous setting.

The angular derivative problem for petals of one-parameter semigroups in the unit disk

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals, we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals, we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy–Wolff point.

ESTIMATE FOR DOMAIN OF UNIVALENCE ON THE CLASS OF HOLOMORPHIC SELF-MAPS OF A DISC WITH TWO BOUNDARY FIXED POINTS

Holomorphic self-maps of the unit disc with two boundary fixed points, one of which is a Denjoy–Wolff point are investigated. An upper estimate for domain of univalence is obtained for functions in such class, which depends on the value of the angular derivative at the repulsive boundary fixed point.

Semigroup-fication of univalent self-maps of the unit disc

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 $(\phi_t)$ of holomorphic self-maps of the unit disc whose time one map $\phi_1$ 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 $\phi_1$. Moreover, in case $f$ (and hence $\phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is almost symmetric with respect to vertical lines.

Denjoy–Wolff theory and spectral properties of weighted composition operators on Hol(D)

We study the spectrum σ(T) of a weighted composition operator T induced by a weight m∈Hol(D) and a holomorphic self-map φ on the unit disc, which is not an elliptic automorphism. If φ has a unique fixed point in D, we show that σ(T) is a bounded discrete set such that σ(T)∖{0} is a set of eigenvalues with multiplicity one. If φ has a Denjoy–Wolff point α on the unit circle, we first prove that the point spectrum is C∖{0} whenever m≠0 is constant. Moreover, the multiplicity of each eigenvalue is infinite. Then we describe classes of m for which the point spectrum of T is either empty or equal to C∖{0}.

Spectra of Weighted Composition Operators with Quadratic Symbols

Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H 2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of Wѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so.

Semigroups of Composition Operators and Integral Operators in BMOA-Type Spaces

The aim of this article is to study semigroups of composition operators \(T_t=f\circ \phi _t\) on the BMOA-type spaces \(\textit{BMOA}_p\), and on their “little oh” analogues \(\textit{VMOA}_p\). The spaces \(\textit{BMOA}_p\) were introduced by R. Zhao as part of the large family of F(p, q, s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces \(D^p_{p-1}\). We study the maximal subspace \([\phi _t, \textit{BMOA}_p]\) of strong continuity, providing a sufficient condition on the infinitesimal generator of \(\{\phi _t\}\), under which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), and a related necessary condition in the case where the Denjoy–Wolff point of the semigroup is in \({{\mathbb {D}}}\). Further, we characterize those semigroups, for which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), in terms of the resolvent operator of the infinitesimal generator of \((T_t|_{\textit{VMOA}_p})\). In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators \(T_g\). We characterize the symbols g for which \(T_g:\,\textit{BMOA}\rightarrow \textit{BMOA}_1\) is bounded or compact, thus extending a related result to the case \(p=1\). We also prove that for \(1<p<2\) compactness of \(T_g\) on \(\textit{BMOA}_p\) is equivalent to weak compactness.

Ergodic Properties of Composition Semigroups on the Disc Algebra

Every semigroup $$\{\varphi _t\}_{t\ge 0}$$ of self-maps of the disc defines a semigroup $$\{C_{\varphi _t}\}_{t\ge 0}$$ of compositions operators on the space of holomorphic functions on the disc. We characterize the (uniform) mean ergodicity (in the sense of continuous means) and the asymptotic behaviour of these operators when they define a $$C_0$$ -semigroup on the disc algebra, in terms of the Denjoy–Wolff point and the associated planar domain in the sense of Berkson and Porta. Finally we deal with the case of Hardy and Bergman spaces.

On the Spectral Value of Semigroups of Holomorphic Functions

Let \((\phi _t)_{t \ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({{\,\mathrm{{\mathbb {D}}}\,}}\) with Denjoy–Wolff point \(\tau =1\). The angular derivative is \(\phi _t^{\prime }(1)= e^{-\lambda t}\), where \(\lambda \ge 0\) is the spectral value of \((\phi _t)\). If \(\lambda >0\) the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset of \({{\,\mathrm{{\mathbb {D}}}\,}}\). We specify the type of the semigroup by examining the asymptotic behavior of \(\phi _t(K)\). We provide a representation of the spectral value of the semigroup with the use of several potential theoretic quantities, e.g., harmonic measure, Green function, extremal length, condenser capacity.

Semigroups of composition operators on [formula omitted] spaces

In this paper we prove that no non-trivial semigroup (φt)t≥0 consisting of analytic self-maps of the unit disk D generates a strongly continuous composition semigroup on Qp for p>0; that is, [φt,Qp]⫋Qp, which answers a question asked by A.G. Siskakis in 1996. We also give an affirmative answer to question in [9] that Qp,0⫋[φt,Qp] for all p>0 if the Denjoy-Wolff point of (φt)t≥0 is on the boundary of D, where Qp,0 is the closure of all polynomials in Qp space. Moreover, for a semigroup (φt)t≥0 with the Denjoy-Wolff point in D, we give a sufficient and necessary condition such that Qp,0⫋[φt,Qp] in terms of the infinitesimal generator of (φt)t≥0 for all p>0. As an application, a new characterization of Qp,0 in terms of semigroup (φt)t≥0 acting on Qp is given, which gives a generalization of the theorem of D. Sarason for VMOA.

On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point

Let \{\phi_t\} be a semigroup of holomorphic self-maps of the unit disc \mathbb{D} with Denjoy–Wolff point \tau\in \partial\mathbb{D} . We study the rate of convergence of the semigroup to \tau , that is, given z\in \overline{\mathbb{D}} , we discuss the behavior of |\phi_{t}(z)-\tau| as t goes to +\infty .

The iterates of composition operators on Banach spaces of holomorphic functions

In this paper we study uniform convergence, strong convergence, weak convergence, and ergodicity of the iterates of composition operators Cφ on various Banach spaces of holomorphic functions on the unit disk, such as Bergman spaces, Dirichlet spaces, weighted Banach spaces with sup-norm, and weighted Bloch spaces. For many spaces, the following results are proved:(i)the iterates Cφn do not converge in the weak operator topology and Cφ is mean ergodic if φ is an elliptic automorphism,(ii)Cφn converge uniformly if the Denjoy-Wolff point of φ is in D,(iii)Cφ is not mean ergodic if the Denjoy-Wolff point of φ lies on the boundary ∂D.

Semigroups of holomorphic functions and condenser capacity

Suppose $$\left( \phi _t \right) _{t \ge 0}$$ is a semigroup of holomorphic functions in the unit disk $$\mathbb {D}$$ with Denjoy–Wolff point $$\tau =1$$. Suppose K is a compact subset of $$\mathbb {D}$$. We prove that the capacity of the condenser $$(\mathbb {D}, \phi _t(K) )$$ is a decreasing function of t. Moreover, we study its asymptotic behavior as $$t \rightarrow + \infty $$ in relation with the type of the semigroup.

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc

Let Δ ⊊ C \Delta \subsetneq \mathbb {C} be a simply connected domain, let f : D → Δ f:\mathbb {D} \to \Delta be a Riemann map, and let { z k } ⊂ Δ \{z_k\}\subset \Delta be a compactly divergent sequence. Using Gromov’s hyperbolicity theory, we show that { f − 1 ( z k ) } \{f^{-1}(z_k)\} converges non-tangentially to a point of ∂ D \partial \mathbb {D} if and only if there exists a simply connected domain U ⊊ C U\subsetneq \mathbb {C} such that Δ ⊂ U \Delta \subset U and Δ \Delta contains a tubular hyperbolic neighborhood of a geodesic of U U and { z k } \{z_k\} is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if ( ϕ t ) (\phi _t) is a non-elliptic semigroup of holomorphic self-maps of D \mathbb {D} with Koenigs function h h and h ( D ) h(\mathbb {D}) contains a vertical Euclidean sector, then ϕ t ( z ) \phi _t(z) converges to the Denjoy-Wolff point non-tangentially for every z ∈ D z\in \mathbb {D} as t → + ∞ t\to +\infty . Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but is oscillating, in the sense that the slope of the trajectories is not a single point.

Asymptotic behavior of orbits of holomorphic semigroups

Let (ϕt) 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 (ϕt). In this paper we characterize the type of convergence of the orbits of (ϕt) 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 axis”. We also prove that such conditions are equivalent to the curve [0,∞)∋t↦ϕt(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.

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.

Normaloid weighted composition operators on H2

When φ is an analytic self-map of the unit disk with Denjoy–Wolff point a∈D, and ρ(Wψ,φ)=ψ(a), we give an exact characterization for when Wψ,φ is normaloid. We also determine the spectral radius, essential spectral radius, and essential norm for a class of non-power-compact composition operators whose symbols have Denjoy–Wolff point in D. When the Denjoy–Wolff point is on ∂D, we give sufficient conditions for several new classes of normaloid weighted composition operators.

Spectra of some weighted composition operators on H2

We completely characterize the spectrum of a weighted composition operator $W_{\psi, \varphi}$ on $H^{2}(\mathbb{D})$ when $\varphi$ has Denjoy-Wolff point $a$ with $0<|\varphi '(a)|< 1$, the iterates, $\varphi_{n}$, converge uniformly to $a$, and $\psi$ is in $H^{\infty}(\mathbb{D})$ and continuous at $a$. We also give bounds and some computations when $|a|=1$ and $\varphi '(a)=1$ and, in addition, show that these symbols include all linear fractional $\varphi$ that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi$ so that $W_{\psi, \varphi}$ is seminormal.

On The Numerical Ranges of Composition Operators Induced By Mappings With The Denjoy-Wolff Point On The Boundary

Let D denote the open unit disk in the complex plane and let \({\phi}\) denote a holomorphic self-map of D. We show that if \({\phi}\) is not linear-fractional and has its Denjoy-Wolff point on the boundary of D, then 0 is an interior point of the numerical range of the composition operator \({C_{\phi}}\) on the Hardy space \({H^2({\mathbf D})}\). This answers a question raised by Bourdon and Shapiro (Integr Equations Oper Theory 44:410–441, 2002), and, together with their results, provides a complete description of when 0 belongs to the numerical range of a composition operator on \({H^2({\mathbf D})}\).

On the rate of convergence of hyperbolic semigroups of holomorphic functions

Let { ϕ t } t ⩾ 0 be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all t > 0 , ϕ t ' ( 1 ) 1 (angular derivative), namely the semigroup is hyperbolic. We prove then that the rate of convergence of the semigroup to the point 1, as t → + ∞ , is exponential with exponent arbitrarily close to - π t / ν ( Ω ) , where ν ( Ω ) is the width of the smallest strip containing the associated planar domain Ω. We also prove that the trajectories of the semigroup approach the point 1 in a monotonic way.