arXiv subjects

Gustavo R. Ferreira

Publications and source records attributed to Gustavo R. Ferreira.

6 recordsLinked to original sources

arXiv2025

Baker domains and orbits disappearing to infinity

We study attracting orbits escaping to infinity in natural families of transcendental entire functions. We show that, if an attracting fixed point escapes to infinity while its multiplier tends to one, then the limiting function has a doubly parabolic Baker domain. Conversely, we show that any function with an invariant doubly parabolic Baker domain can be approximated locally uniformly by functions in its quasiconformal equivalence class having an attracting fixed point whose multiplier tends to one.

math.DS↗

arXiv2025

Teichmüller spaces and normal forms associated to wandering domains

We study the dynamical Teichmüller space ${\mathcal T}(U,f)$ associated to a wandering domain $U$ of an entire function $f$. We show that a discrete grand orbit relation in $U$ forces ${\mathcal T}(U,f)$ to be infinite dimensional, thereby answering a question of Fagella--Henriksen. We further describe the geometry of these spaces by developing normal forms for the dynamics on wandering domains, yielding global linearising coordinates in the discrete case and power-type dynamics between annuli in the indiscrete case.

math.DS↗

arXiv2025

Boundaries of multiply connected Fatou components. A unified approach

We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any type) as well as wandering domains. We prove that any Fatou component admits a harmonic measure on its boundary whose support is the whole boundary. Consequently, we relate, in a successful way, the geometric structure of such Fatou components (in terms of the limit sets of their universal covering maps), the dynamics induced on their boundary from an ergodic point of view, and analytic properties of the associated inner function.

math.DS↗

arXiv2025

Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich's result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set $M_f$ of entire functions quasiconformally equivalent to $f$ admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains -- another hot topic of research in complex dynamics. Given an entire function $f$ with a simply connected wandering domain $U$, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as $f$ moves within appropriate parameter families.

math.DS↗

arXiv2024

Boundary behaviour of universal covering maps

Let $Ω\subset\widehat{\mathbb{C}}$ be a multiply connected domain, and let $π\colon \mathbb{D}\toΩ$ be a universal covering map. In this paper, we analyze the boundary behaviour of $π$, describing the interplay between radial limits and angular cluster sets, the tangential and non-tangential limit sets of the deck transformation group, and the geometry and the topology of the boundary of $Ω$. As an application, we describe accesses to the boundary of $Ω$ in terms of radial limits of points in the unit circle, establishing a correspondence in the same spirit as in the simply connected case. We also develop a theory of prime ends for multiply connected domains which behaves properly under the universal covering, providing an extension of the Carathéodory--Torhorst Theorem to multiply connected domains.

math.CV↗

arXiv2018

Principal Component Analysis: A Natural Approach to Data Exploration

Principal component analysis (PCA) is often used for analyzing data in the most diverse areas. In this work, we report an integrated approach to several theoretical and practical aspects of PCA. We start by providing, in an intuitive and accessible manner, the basic principles underlying PCA and its applications. Next, we present a systematic, though no exclusive, survey of some representative works illustrating the potential of PCA applications to a wide range of areas. An experimental investigation of the ability of PCA for variance explanation and dimensionality reduction is also developed, which confirms the efficacy of PCA and also shows that standardizing or not the original data can have important effects on the obtained results. Overall, we believe the several covered issues can assist researchers from the most diverse areas in using and interpreting PCA.

cs.CE↗