arXiv subjects

Manuel D. Contreras

Publications and source records attributed to Manuel D. Contreras.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step

The study of holomorphic self-mappings of the unit disc commuting under the composition goes back to A.L. Shields (1964), W.A. Pranger (1970), D.F. Behan (1973), and C.C. Cowen (1984). In many situations, the centralizer of a holomorphic self-map ${φ:\mathbb D \to \mathbb D}$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ\in\mathsf{Hol}(\mathbb D):ψ\circφ=φ\circψ\}$ turns out to be commutative. However, this does not hold for the case of a parabolic self-map $φ$ of positive hyperbolic step, which is analyzed in detail in this paper. We investigate the relationships among commutativity, simultaneous linearization, and holomorphic models. In particular, we obtain existence and uniqueness results for the simultaneous linearization of commuting pairs $φ$, $ψ\in \mathcal Z_\forall(φ)$. Furthermore, extending this notion to arbitrary families of holomorphic self-mappings, we show that a given (finite or infinite) family in the centralizer ${Δ\subset\mathcal Z_\forall(φ)}$ can be simultaneously linearized together with$~φ$ if and only if any two elements of$~Δ$ commute with each other. This gives a far reaching extension of Cowen's result concerning commuting pairs in$~\mathsf{Hol}(\mathbb D)$.

math.CV↗

arXiv2026

The Bergman and the growth numbers of domains and hyperbolic geometry

In this article we completely characterize the Bergman number of general domains. Namely, we prove that the Bergman number of a hyperbolic unbounded domain can be calculated in terms of the asymptotic behavior of its hyperbolic metric near infinity. To obtain the proof of this result we first work in the setting of growth spaces, defining the growth number of a domain. Later, we prove an equality relating the Bergman number and the growth number of domains. We also provide examples of domains with prescribed Bergman number which have zero Hardy number, solving a question posed previously by Betsakos and Cruz-Zamorano. At the end, a similar idea is treated for the case of Bloch-type spaces.

math.CV↗

arXiv2025

Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step

Let $φ:\mathbb D \to \mathbb D$ be a parabolic self-map of the unit disc $\mathbb D$ having zero hyperbolic step. We study holomorphic self-maps of $\mathbb D$ commuting with $φ$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $ψ\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $φ$ if and only if the two self-maps have the same Denjoy-Wolff point and $ψ$ is a pseudo-iterate of $φ$ in the sense of Cowen. Moreover, we show that the centralizer of $φ$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ:ψ\circφ=φ\circψ\}$ is commutative. We also prove that if $φ$ is univalent, then all elements of $\mathcal Z_\forall(φ)$ are univalent as well, and if $φ$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(φ)$. The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.

math.CV↗

arXiv2024

Examples in Discrete Iteration of Arbitrary Intervals of Slopes

Given a compact interval $[a,b] \subset [0,π]$, we construct a parabolic self-map of the upper half-plane whose set of slopes is $[a,b]$. The nature of this construction is completely discrete and explicit: we explicitly construct a self-map and we explicitly show in which way its orbits wander towards the Denjoy-Wolff point. We also analyze some properties of the Herglotz measure corresponding to such example, which yield the regularity of such self-map in its Denjoy-Wolff point.

math.CV↗

arXiv2024

On the Hardy number of Koenigs domains

This work studies the Hardy number for the class of hyperbolic planar domains satisfying Abel's inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that for all regular domains in the above class, the Hardy number is greater or equal than $1/2$, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.

math.CV↗

arXiv2024

Hardy Number of Koenigs Domains: Sharp Estimate

Let $Ω$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $Ω$ is greater or equal to $1/2$. That is, every holomorphic function in the unit disc $f \colon \mathbb{D} \to Ω$ belongs to the Hardy space $H^{p}(\mathbb{D})$ for all $p<1/2$.

math.CV↗

arXiv2024

The Slope Problem in Discrete Iteration

The slope problem in holomorphic dynamics in the unit disk goes back to Wolff in 1929. However, there have been several contributions to this problem in the last decade. In this article the problem is revisited, comparing the discrete and continuous cases. Some advances are derived in the discrete parabolic case of zero hyperbolic step, showing that the set of slopes has to be a closed interval which is independent of the initial point. The continuous setting is used to show that any such interval is a possible example. In addition, the set of slopes of a family of parabolic function is discussed, leading to examples of functions with some regularity whose set of slopes is non-trivial.

math.CV↗

arXiv2024

Characterization of the hyperbolic step of parabolic functions

A classical problem in Complex Dynamics on hyperbolic domains is to characterize the hyperbolic step of parabolic functions. This topic has been studied by several authors, leading to different results and providing characterizations that depend on the behaviour of the iterates of such function. In this work we provide new characterizations in terms of intrinsic properties of the function.

math.CV↗

arXiv2024

Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc

Let $φ$ be a univalent non-elliptic self-map of the unit disc $\mathbb D$ and let $(ψ_{t})$ be a continuous one-parameter semigroup of holomorphic functions in $\mathbb D$ such that $ψ_{1}\neq\mathrm{id}_{\mathbb D}$ commutes with $φ$. This assumption does not imply that all elements of the semigroup $(ψ_t)$ commute with $φ$. In this paper, we provide a number of sufficient conditions that guarantee that ${ψ_t\circφ=φ\circψ_t}$ for all ${t>0}$: this holds, for example, if $φ$ and $ψ_1$ have a common boundary (regular or irregular) fixed point different from their common Denjoy-Wolff point $τ$, or when $ψ_1$ has a boundary regular fixed point ${σ\neqτ}$ at which $φ$ is isogonal, or when $(φ-\mathrm{id}_{\mathbb D})/(ψ_1-\mathrm{id}_{\mathbb D})$ has an unrestricted limit at $τ$. In addition, we analyze how $φ$ behaves in the petals of the semigroup $(ψ_t)$.

math.CV↗

arXiv2024

On the uniform convergence of continuous semigroups

Let $Ω$ be a region in the complex plane $\mathbb C$ and let $\{Φ_t \}_{t\ge 0}$ be a continuous semigroup of functions on $Ω$; that is, $Φ_t:Ω\toΩ$ is holomorphic for every $t\ge 0$, $Φ_0(z)=z$, for every $z\inΩ$, $Φ_t\circΦ_s=Φ_{s+t}$, for every $s$, $t\ge 0$, and \[ Φ_t(z)\to z\,,\quad t\to0^+, \] uniformly on compact subsets of $Ω$. Despite this definition only requires the uniform convergence on compact subsets, P. Gumenyuk proved in 2014 that, when $Ω$ is the unit disc, the convergence is uniform on the whole $\mathbb D$. In this paper, we enhance Gumenyuk's result by proving that for every continuous semigroup $\{Φ_t\}_{t\ge 0}$ on $\mathbb D$ we have $$ \sup_{z\in\mathbb D}|Φ_t(z)-z|= O(\sqrt t), \ t\to0^+. $$ In addition, we provide an example showing that $O(\sqrt t)$ is the best possible rate of uniform convergence valid for all semigroups on $\mathbb D$. When $Ω$ is the right half-plane $\mathbb C_+$, we consider semigroups $\{Φ_t\}$ with $\infty$ as its Denjoy-Wolff point. It is not difficult to show that Gumenyuk's result is no longer true for these semigroups. Our second result characterises when such continuous semigroups converge uniformly to the identity, as $t\to0^+$, in terms of their infinitesimal generators. Namely, this convergence holds if and only if the infinitesimal generator of the semigroup is bounded in the half-plane $\{z\in \mathbb C:\, \Re z>1\}$. In this case, we can also prove that the rate of convergence is again $O(\sqrt{t})$, as $t\to0^+$. An example of application of this result is when the semigroup is in the Gordon-Hedenmalm class (the one producing bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is harmonic measure, which we have done through a classic result of M. Lavrentiev.

math.CV↗

arXiv2024

Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map $φ$ of the unit disc $\mathbb D$ in order to be contained in a continuous one-parameter semigroup. In this paper, we tackle this embedding problem by establishing different dichotomy results about the centralizer of $φ$ (i.e. the set of all univalent self-maps commuting with $φ$) which depend strongly on the dynamical character of $φ$. Our approach is, in part, based on a new technique to obtain simultaneous linearizations of two non-elliptic univalent self-maps of the unit disc, which might be interesting on their own. We also introduce and study several closed additive subsemigroups of the complex plane that collect the main features of the centralizer of $φ$ and which play a prominent position in those dichotomy results.

math.CV↗

arXiv2023

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series $\mathcal{A}(\mathcal{C}_{+})$ consists on those Dirichlet series convergent in the right half-plane $\mathcal{C}_{+}$ and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols $Φ:\mathcal{C}_{+}\to\mathcal{C}_{+}$ giving rise to bounded composition operators $\mathit{C}_Φ$ in $\mathcal{A}(\mathcal{C}_{+})$ and denote this class by $\mathcal{G}_{\mathcal{A}}$. We also characterise when the operator $\mathit{C}_Φ$ is compact in $\mathcal{A}(\mathit{C}_{+})$. As a byproduct, we show that the weak compactness is equivalent to the compactness for $\mathit{C}_Φ$. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions $\{Φ_{t}\}$ in the class $\mathcal{G}_{\mathcal{A}}$ and strongly continuous semigroups of composition operators $\{T_{t}\}$, $T_{t}f=f\circΦ_{t}$, $f\in\mathcal{A}(\mathcal{C}_{+})$. We conclude providing examples showing the differences between the symbols of bounded composition operators in $\mathcal{A}(\mathcal{C}_{+})$ and the Hardy spaces of Dirichlet series $\mathcal{H}^{p}$ and $\mathcal{H}^{\infty}$.

math.FA↗

arXiv2022

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{Φ_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circΦ_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\in[1,\infty)$. For the case $p=\infty$, we prove that there is no non-trivial strongly continuous semigroup of composition operators in $\mathcal{H}^\infty$. We characterize the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as those Dirichlet series sending $\mathbb C_{+}$ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

math.FA↗

arXiv2020

Angular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk

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.

math.CV↗

arXiv2020

Integration operators in average radial integrability spaces of analytic functions

In this paper we characterize the boundedness, compactness, and weak compactness of the integration operators \begin{align*} T_g (f)(z)=\int_{0}^{z} f(w)g'(w)\ dw \end{align*} acting on the average radial integrability spaces $RM(p,q)$. For these purposes, we develop different tools such as a description of the bidual of $RM(p,0)$ and estimates of the norm of these spaces using the derivative of the functions, a family of results that we call Littlewood-Paley type inequalities.

math.FA↗

arXiv2020

Infinitesimal generators of semigroups with prescribed boundary fixed points

We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb D$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning.vWe also give a new proof of the classical Cowen-Pommerenke inequalities for univalent self-maps of $\mathbb D$.

math.CV↗

arXiv2020

Average radial integrability spaces of analytic functions

In this paper we introduce the family of spaces $RM(p,q)$, $1\leq p,q\leq +\infty$. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when $p=\infty$) and Bergman spaces (when $p=q$). We characterize the inclusion between $RM(p_1,q_1)$ and $RM(p_2,q_2)$ depending on the parameters. For $1<p,q<\infty$, our main result provides a characterization of the dual spaces of $RM(p,q)$ by means of the boundedness of the Bergman projection. We show that $RM(p,q)$ is separable if and only if $q<+\infty$. In fact, we provide a method to build isomorphic copies of $\ell^\infty$ in $RM(p,\infty)$.

math.FA↗

arXiv2019

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

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 if $(ϕ_t)$ is a non-elliptic semigroup of holomorphic self-maps of $\mathbb D$ with Königs function $h$ and $h(\mathbb D)$ contains a vertical Euclidean sector, then $ϕ_t(z)$ converges to the Denjoy-Wolff point non-tangentially for every $z\in \mathbb D$ as $t\to +\infty$. 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 oscillating, in the sense that the slope of the trajectories is not a single point.

math.CV↗