arXiv subjects

David Martí-Pete

Publications and source records attributed to David Martí-Pete.

11 recordsLinked to original sources

arXiv2026

A counterexample to Eremenko's conjecture

The escaping set of a transcendental entire function consists of those points that tend to infinity under iteration. Eremenko asked whether every connected component of this set is unbounded. In arXiv:2108.10256, we constructed transcendental entire functions for which the escaping set has prescribed compact connected components; in particular, it may have a singleton component. This disproves the conjecture. In this note, we give an introduction to the problem, aimed at a general mathematical audience. We also sketch a variant of the construction in which a prescribed half-strip is an oscillating wandering domain and an escaping point lies on its boundary. The account is based on the second author's lecture at the 2026 International Congress of Basic Science.

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arXiv2026

Wandering dynamics of transcendental functions

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $φ$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $φ$ and all its iterates $φ^n$ are defined on $K$, and $φ^n(K)\to\infty$ as $n\to\infty$. We prove that there exist a transcendental meromorphic function $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ and a compact set $\widetilde{K}$ such that the dynamics of $f$ on the orbit of $\widetilde{K}$ is conjugate, via a smooth change of coordinate close to the identity, to that of $φ$ on the orbit of $K$. If $K$ does not separate the plane, the function $f$ may be chosen to be entire. If all iterates of $φ$ are univalent on $K$, we can take $\widetilde{K}=K$. We also prove a similar theorem for oscillating dynamics. Finally, we use our results to answer a number of questions of Benini et al. concerning wandering domains of entire functions.

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arXiv2025

On the dimension of the boundaries of attracting basins of entire maps

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $ζ$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the attracting basin of (the orbit of) $ζ$ have hyperbolic (and, consequently, Hausdorff) dimension larger than $1$, provided $f^p$ has an infinite degree on an immediate component $U$ of the basin, and the singular set of $f^p|_U$ is compactly contained in $U$. The same holds for the boundaries of components of the basin of a parabolic $p$-periodic point $ζ$, under the additional assumption $ζ\notin \overline{\text{Sing}(f^p)}$. We also prove that if an immediate component of an attracting basin of an arbitrary transcendental entire map is bounded, then the boundaries of components of the basin have hyperbolic dimension larger than $1$. This enables us to show that the boundary of a component of an attracting basin of a transcendental entire function is never a smooth or rectifiable curve. The results provide a partial answer to a question from Hayman's list of problems in function theory.

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arXiv2024

Eremenko's conjecture, wandering Lakes of Wada, and maverick points

We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems. (1) We construct a counterexample to Eremenko's conjecture, a central problem in transcendental dynamics that asks whether every connected component of the set of escaping points of a transcendental entire function is unbounded. (2) We prove that there is a transcendental entire function for which infinitely many Fatou components share the same boundary. This resolves the long-standing problem whether "Lakes of Wada continua" can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. (3) We answer a question of Rippon concerning the existence of non-escaping points on the boundary of a bounded escaping wandering domain, that is, a wandering Fatou component contained in the escaping set. In fact we show that the set of such points can have positive Lebesgue measure. (4) We give the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a question of Boc Thaler. In view of (3), we introduce the concept of "maverick points": points on the boundary of a wandering domain whose accumulation behaviour differs from that of internal points. We prove that the set of such points has harmonic measure zero, but that it can nonetheless be rather large. For example, it may have positive planar Lebesgue measure.

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arXiv2023

Bounded Fatou and Julia components of meromorphic functions

We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.

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arXiv2019

Spiders' webs in the punctured plane

Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in $\mathbb{C}$. We show that there are many transcendental self-maps of $\mathbb{C}^*$ for which the Julia set is such a spider's web, and we construct a transcendental self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this structure and hence is connected. By way of contrast with transcendental entire functions, we conjecture that there is no transcendental self-map of $\mathbb{C}^*$ for which the fast escaping set $A(f)$ is such a spider's web.

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arXiv2019

On the connectivity of the escaping set in the punctured plane

We consider the dynamics of transcendental self-maps of the punctured plane, $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$. We prove that the escaping set $I(f)$ is either connected, or has infinitely many components. We also show that $I(f)\cup \{0,\infty\}$ is either connected, or has exactly two components, one containing $0$ and the other $\infty$. This gives a trichotomy regarding the connectivity of the sets $I(f)$ and $I(f)\cup \{0,\infty\}$, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in $\mathbb{C}^*$ by constructing the first such example. We also prove that if $f$ has a doubly connected Baker domain, then its closure contains both $0$ and $\infty$, and hence $I(f)\cup\{0,\infty\}$ is connected.

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arXiv2018

Wandering domains for entire functions of finite order in the Eremenko-Lyubich class

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.

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arXiv2017

Escaping Fatou components of transcendental self-maps of the punctured plane

We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

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arXiv2016

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$ under iteration by $f$.

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arXiv2014

The escaping set of transcendental self-maps of the punctured plane

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb C^*=\mathbb C\setminus\{ 0\}$ for which both 0 and $\infty$ are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either 0, $\infty$ or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points each of which has the Julia set as its boundary.

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