arXiv subjects

Francisco J. Cruz-Zamorano

Publications and source records attributed to Francisco J. Cruz-Zamorano.

12 recordsLinked to original sources

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

Extremal rate of convergence in continuous dynamics

This paper deals with semigroups of holomorphic self-maps of the upper half-plane that exhibit an extremal (i.e. the slowest possible) rate of convergence to their Denjoy--Wolff point. The main novelty lies in the parabolic case of zero hyperbolic step. We provide several characterizations for such semigroups in terms of the Herglotz representation of their infinitesimal generators, the conformality at the Denjoy--Wolff point of a modification of their associated Koenigs function, and more.

math.CV↗

arXiv2025

Extremal rate of convergence in discrete hyperbolic and parabolic dynamics

This paper investigates the dynamical behaviour of holomorphic self-maps of the upper half-plane. More precisely, we focus on the hyperbolic and parabolic self-maps whose orbits approach the Denjoy--Wolff point with the slowest possible rate. We characterize self-maps of such extremal rate using various tools, like the Herglotz representation, the conformality of the Koenigs function at the Denjoy--Wolff point and the hyperbolic distance.

math.CV↗

arXiv2025

On the Hardy number and the Bergman number of a planar domain

This article deals with functions with a prefixed range and their inclusions in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. We provide a new characterization for the Hardy number of a domain in terms of its Green function. Based on this, we present a class of domains for which the Hardy number and the Bergman number coincide. However, in general, we show that the Hardy number and the Bergman number of a domain are not equal; even for domains which are regular for the Dirichlet problem.

math.CV↗

arXiv2025

On the rates of convergence of orbits in semigroups of holomorphic functions

Let $(ϕ_t)$ be a continuous semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $τ\in\overline{\mathbb{D}}$. We study the rate of convergence of the forward orbits of $(ϕ_t)$ to the Denjoy-Wolff point by finding explicit bounds for the quantity $|ϕ_t(z)-τ|$, $z\in\overline{\mathbb{D}}$, $t > 0$. We further discuss the corresponding rate of convergence for the backward orbits of $(ϕ_t)$.

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

The Hardy number and the Bergman number of a planar domain are equal

This article deals with functions with a prefixed range and their inclusion in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. In particular, we improve a theorem of Karafyllia showing that the Hardy and Bergman numbers of any given domain coincide, that is, the Hardy and weighted Bergman spaces to which a function with prefixed range belongs can be related. The main tools in the proofs are the Green function of the domain and its universal covering map.

math.CV↗

arXiv2024

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.

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↗