arXiv subjects

Núria Fagella

Publications and source records attributed to Núria Fagella.

At least 19 recordsLinked to original sources

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.

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arXiv2025

From pathological to paradigmatic: A retrospective on Eremenko and Lyubich's entire functions

This article surveys the impact of Eremenko and Lyubich's paper ''Examples of entire functions with pathological dynamics'', published in 1987 in the Journal of the LMS. Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics.

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arXiv2024

On the parameter space of fibered hyperbolic polynomials

We present an application of quasiconformal (QC) surgery for holomorphic maps fibered over an irrational rotation of the unit circle, also known as quasiperiodically forced (QPF) maps. It consists of modifying the fibered multiplier of an attracting invariant curve for a QPF hyperbolic polynomial. This is the analogue of the classical change of multiplier of an attracting cycle in the one-dimensional iteration case, to parametrize hyperbolic components of the Mandelbrot set. Our goal is to show that, for a family of QPF quadratic maps with Diophantine frequency, the fibered multiplier map associated to its unique attracting invariant curve, as a complex counterpart of the Lyapunov exponent, is a holomorphic submersion on a complex Banach manifold.

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arXiv2024

Boundary dynamics in unbounded Fatou components

We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected. Hence, by means of a Riemann map $ φ\colon\mathbb{D}\to U $ and the associated inner function, the boundary of $ U $ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $ \mathbb{C} $. Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $ \partial U $, being all periodic boundary points accessible from $ U $. Finally, under the same conditions, the set of singularities of $ g $ is shown to have zero Lebesgue measure.

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arXiv2024

Local connectivity of boundaries of tame Fatou components of meromorphic functions

We prove local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class $\mathcal B$, and the boundaries of the basins (possibly unbounded) are allowed to contain an infinite number of post-singular values, as well as the essential singularity at infinity. A basic assumption is that the unbounded parts of the basins are contained in regions which we call `repelling petals at infinity', where the map exhibits a kind of `parabolic' behaviour. In particular, our results apply to a wide class of Newton's methods for transcendental entire maps. As an application, we prove local connectivity of the Julia set of Newton's method for $\sin z$, providing the first non-trivial example of a locally connected Julia set of a transcendental map outside class $\mathcal B$, with an infinite number of unbounded Fatou components.

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arXiv2024

Grand orbit relations in wandering domains

One of the fundamental distinctions in McMullen and Sullivan's description of the Teichmüller space of a complex dynamical system is between discrete and indiscrete grand orbit relations. We investigate these on the Fatou set of transcendental entire maps and provide criteria to distinguish between the two types. Furthermore, we show that discrete and indiscrete grand orbit relations may coexist non-trivially in a wandering domain, a phenomenon which does not occur for any other type of Fatou component. One of the tools used is a novel quasiconformal surgery technique of independent interest.

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arXiv2024

Shrinking targets and recurrent behaviour for forward compositions of inner functions

We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the simpler case of iteration. A result of Fernández, Melián and Pestana gives a precise version of the classical Poincaré recurrence theorem for iterates of the boundary extension of an inner function that fixes~0. We generalise this to forward composition sequences $F_n=f_n\circ \dots\circ f_1,$ $n\in \mathbb{N},$ where $f_n$ are inner functions that fix~0, giving conditions on the contraction of $(F_n)$ so that the radial boundary extension $F_n$ hits any shrinking target of arcs $(I_n)$ of a given size. Next, Aaronson, and also Doering and Mañé, gave a remarkable dichotomy for iterates of any inner function, showing that the behaviour of the boundary extension is of two entirely different types, depending on the size of the sequence $(|f^n(0)|)$. In earlier work, we showed that one part of this dichotomy holds in the non-autonomous setting of forward compositions. It turns out that this dichotomy is closely related to the result of Fernández, Melián and Pestana, and here we show that a version of the second part of the dichotomy holds in the non-autonomous setting provided we impose a condition on the contraction of $(F_n)$ in relation to the size of the sequence $(|F_n(0)|)$. The techniques we use include a strong version of the second Borel--Cantelli lemma and strong mixing results of Pommerenke for contracting sequences of inner functions. We give examples to show that the contraction conditions that we need to impose in the non-autonomous setting are best possible.

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arXiv2023

Bifurcation loci of families of finite type meromorphic maps

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of Mañé-Sad-Sullivan for rational maps of the Riemann sphere and those of Eremenko and Lyubich for entire maps of finite type of the complex plane, and essentially closes the problem of density of structural stability for holomorphic dynamical systems in one complex variable with finitely many singular values. This result is obtained as a consequence of a detailed study of a new type of bifurcation that arises with the presence of both poles and essential singularities (namely periodic orbits exiting the domain of definition of the map along a parameter curve), and in particular its relation with the bifurcations in the dynamics of singular values. The presence of these new bifurcation parameters require essentially different methods to those used in previous work for rational or entire maps.

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arXiv2023

Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps

We present a one-parameter family $F_λ$ of transcendental entire functions with zeros, whose Newton's method yields wandering domains, coexisting with the basins of the roots of $F_λ$. Wandering domains for Newton maps of zero-free functions have been built before by, e.g., Buff and Rückert based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions $f(z)$ in the complex plane that are semiconjugate, via the exponential, to some map $g(w)$, which may have at most a countable number of essential singularities. In this paper we make a systematic study of the general relation (dynamical and otherwise) between $f$ and $g$, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those $g$ of finite-type. We apply these results to characterize the entire functions with zeros whose Newton's method projects to some map $g$ which is defined at both $0$ and $\infty$. The family $F_λ$ is the simplest in this class, and its parameter space shows open sets of $λ$-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton's root-finding method fails.

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arXiv2023

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the first time, we address such questions in the very general setting of sequences $(F_n)$ of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. Some of our results are new even in the classical setting. Our methods apply in particular to non-autonomous dynamical systems, when $(F_n)$ are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics. The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a `weak independence' version of the second Borel--Cantelli lemma and the concept from ergodic theory of `shrinking targets'.

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arXiv2023

A model for boundary dynamics of Baker domains

We consider the transcendental entire function $ f(z)=z+e^{-z} $, which has a doubly parabolic Baker domain $U$ of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in $\partial U$ whose orbit escapes to infity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, it is organized in curves encoded by some symbolic dynamics, whose closure is precisely $\partial U$. We also prove that nevertheless, all escaping points in $\partial U$ are non-accessible from $U$, as opposed to points in $\partial U$ having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in @U, answering a question posted Baranski, Fagella, Jarque and Karpinska. None of these features are known to occur for a general doubly parabolic Baker domain.

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arXiv2021

Dynamical mechanism behind ghosts unveiled in a map complexification

Complex systems such as ecosystems, electronic circuits, lasers or chemical reactions can be modelled by dynamical systems which typically experience bifurcations. Transients typically suffer extremely long delays at the vicinity of bifurcations and it is also known that these transients follow scaling laws as the bifurcation parameter gets closer the bifurcation value in deterministic systems. The mechanisms involved in local bifurcations are well-known. However, for saddle-node bifurcations, the relevant dynamics after the bifurcation occur in the complex phase space. Hence, the mechanism responsible for the delays and the associated inverse-square root scaling law for this bifurcation can be better understood by looking at the dynamics in the complex space. We follow this approach and complexify a simple ecological system undergoing a saddle-node bifurcation. The discrete model describes a biological system with facilitation (cooperation) under habitat destruction for species with non-overlapping generations. We study the complex (as opposed to real) dynamics once the bifurcation has occurred. We identify the fundamental mechanism causing these long delays (called ghosts), given by two repellers in the complex space. Such repellers appear to be extremely close to the real line, thus forming a narrow channel close to the two new fixed points and responsible for the slow passage of the orbits, which remains tangible in the real numbers phase space. We analytically provide the relation between the inverse square-root scaling law and the multipliers of these repellers. We finally prove that the same phenomenon occurs for more general i.e., non-necessarily polynomial, models.

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arXiv2019

Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

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arXiv2019

A bound on the number of rationally invisible repelling orbits

We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are $q<\infty$ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by $q$. In particular, there are at most $q$ rationally invisible repelling periodic orbits. The techniques presented here also apply to the more general setting in which the function is allowed to have infinitely many singular values.

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arXiv2019

Univalent Wandering Domains in the Eremenko-Lyubich Class

We use the folding theorem of Bishop to construct an entire function $f$ in class $B$ and a wandering domain $U$ of $f$ such that $f$ restricted to $f^n(U)$ is univalent, for all $n \geq 0$. The components of the wandering orbit are bounded and surrounded by the postcritical set.

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arXiv2018

Singularities of inner functions associated with hyperbolic maps

Let $f$ be a function in the Eremenko-Lyubich class $\mathcal{B}$, and let $U$ be an unbounded, forward invariant Fatou component of $f$. We relate the number of singularities of an inner function associated to $f|_U$ with the number of tracts of $f$. In particular, we show that if $f$ lies in either of two large classes of functions in $\mathcal{B}$, and also has finitely many tracts, then the number of singularities of an associated inner function is at most equal to the number of tracts of $f$. Our results imply that for hyperbolic functions of finite order there is an upper bound -- related to the order -- on the number of singularities of an associated inner function.

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arXiv2017

Singular values and non-repelling cycles for entire transcendental maps

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.

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arXiv2017

Fatou components and singularities of meromorphic functions

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that ${\rm dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\mathbb C$) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

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