arXiv subjects

Lasse Rempe

Publications and source records attributed to Lasse Rempe.

At least 19 recordsLinked to original sources

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 $\varphi$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $\varphi$ and all its iterates $\varphi^n$ are defined on $K$, and $\varphi^n(K)\to\infty$ 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 $\varphi$ 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 $\varphi$ 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

The escaping set in transcendental dynamics

The escaping set of an entire function consists of the points in the complex plane that tend to infinity under iteration. This set plays a central role in the dynamics of transcendental entire functions. The goal of this survey is to explain this role, to summarise some of the main results in the area, and to identify a number of open questions.

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arXiv2024

Spiders' webs in the Eremenko-Lyubich class

Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture of Sixsmith from 2020. In fact, we show that the "fast escaping set", i.e. the set of points whose orbits tend to infinity at an iterated exponential rate, is a spider's web. This answers a question of Rippon and Stallard from 2012. We also discuss a wider class of functions to which our results apply, and state some open questions.

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arXiv2024

Classifying multiply connected wandering domains

We study the internal dynamics of multiply connected wandering domains of meromorphic functions. We do so by considering the sequence of injectivity radii along the orbit of a base point, together with the hyperbolic distortions along the same orbit. The latter sequence has previously been used in this context; the former introduces geometric information about the shape of the wandering domains that interacts with the dynamical information given by the hyperbolic distortions. Using this idea, we complete the description of the internal dynamics of any wandering domain of a meromorphic function, and also unify previous approaches to the question. We conclude that the internal dynamics of a wandering domain, from the point of view of hyperbolic geometry, can be classified into six different types. Five of these types were previously known and are realised by wandering domains of entire functions. The sixth type arises only for meromorphic functions: a locally but not globally eventually isometric wandering domain. We construct a meromorphic function with such a domain, demonstrating that this new phenomenon does in fact occur. Our results show that, on a local level, the dynamical behaviour of wandering domains of meromorphic functions is similar to that which occurs of entire functions. However, new global phenomena can arise in the non-entire case. In particular, we identify a category of multiply connected wandering domains~-- essentially thin and infinitesimally non-contracting domains~-- that naturally generalise multiply connected wandering domains of entire functions, and share many characteristics with these. We generalise a number of results of Bergweiler, Rippon and Stallard for the entire case to this more general setting. In particular, we show the existence of dynamically meaningful singular foliations provided by level lines of certain positive harmonic functions.

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arXiv2023

Points of convergence -- music meets mathematics

"Phase-locking" is a fundamental phenomenon in which coupled or periodically forced oscillators synchronise. The Arnold family of circle maps, which describes a forced oscillator, is the simplest mathematical model of phase-locking and has been studied intensively since its introduction in the 1960s. The family exhibits regions of parameter space where phase-locking phenomena can be observed. A long-standing question asked whether "hyperbolic" parameters~-- those whose behaviour is dominated by periodic attractors, and which are therefore stable under perturbation~-- are dense within the family. A positive answer was given in 2015 by van Strien and the author, which implies that, no matter how chaotic a map within the family may behave, there are always systems with stable behaviour nearby. This research was a focal point of a pioneering collaboration with composer Emily Howard, commencing with Howard's residency in Liverpool's mathematics department in 2015. The collaboration generated impacts on creativity, culture and society, including several musical works by Howard, and lasting influence on artistic practice through a first-of-its-kind centre for science and music. We describe the research and the collaboration, and reflect on the factors that contributed to the latter's success.

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arXiv2022

Entire functions with Cantor bouquet Julia sets

A hyperbolic transcendental entire function with connected Fatou set is said to be of disjoint type. It is known that the Julia set of a disjoint-type function of finite order is a Cantor bouquet; in particular, it is a collection of arcs (''hairs''), each connecting a finite endpoint to infinity. We show that the latter property is equivalent to the function being criniferous (a necessary condition for having a Cantor bouquet Julia set). On the other hand, we show that there is a criniferous disjoint-type entire function whose Julia set is not a Cantor bouquet. We also provide a new characterisation of Cantor bouquet Julia sets in terms of the existence of certain absorbing sets for the set of escaping points, and use this to give a new intrinsic description of a class of entire functions previously introduced by the first author. Finally, the main known sufficient condition for Cantor bouquet Julia sets is the so-called head-start condition of Rottenfusser et al. Under a mild geometric assumption, we prove that this condition is also necessary.

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arXiv2022

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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arXiv2022

Second order linear differential equations with a basis of solutions having only real zeros

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

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arXiv2021

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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arXiv2021

A bouquet of pseudo-arcs

We prove the existence of a transcendental entire function whose Julia set is a "bouquet of pseudo-arcs". More precisely, the union of the Julia set with infinity is an uncountable union of pseudo-arcs, which are pairwise disjoint except at infinity. The existence of such a function follows from a more general result of the second author, but our construction is considerably simpler and more explicit. In particular, the function we construct can be chosen to have lower order $1/2$, while the lower order in the previously known example is infinite.

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arXiv2021

The Eremenko-Lyubich constant

Eremenko and Lyubich proved that an entire function whose set of singular values is bounded is expanding at points where its image has large modulus. These expansion properties have been at the centre of the subsequent study of this class of functions, now called the Eremenko-Lyubich class. We improve the estimate of Eremenko and Lyubich, and show that the new estimate is asymptotically optimal. As a corollary, we obtain an elementary proof that functions in the Eremenko-Lyubich class have lower order at least $1/2$.

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arXiv2021

Non-compact Riemann surfaces are equilaterally triangulable

We show that every open Riemann surface can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, it is a _Belyi surface_: There exists a holomorphic branched covering to the Riemann sphere that is branched only over three values. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

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arXiv2021

Escaping sets are not sigma-compact

Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $σ$-compact, resolving a question of Rippon from 2009.

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arXiv2020

Singular orbits and Baker domains

We show that there is a transcendental meromorphic function with an invariant Baker domain $U$ such that every singular value of $f$ is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that $U$ can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljevi\'c and the author from 2013, and complements recent work of Bara\'nski et al concerning this question.

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arXiv2020

Fatou's associates

Suppose that $f$ is a transcendental entire function, $V \subsetneq \mathbb{C}$ is a simply connected domain, and $U$ is a connected component of $f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to see that $g$ is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in $V$ lie away from the boundary, there is a strong relationship between singularities of $g$ and accesses to infinity in $U$. In the case where $U$ is a forward-invariant Fatou component of $f$, this leads to a very significant generalisation of earlier results on the number of singularities of the map $g$. If $U$ is a forward-invariant Fatou component of $f$ there are currently very few examples where the relationship between the pair $(f, U)$ and the function $g$ have been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this way, and we show the following: For every finite Blaschke product $g$ whose Julia set coincides with the unit circle, there exists a transcendental entire function $f$ with an invariant Fatou component such that $g$ is associated to $f$ in the above sense. Furthermore, there exists a single transcendental entire function $f$ with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated to the restriction of $f$ to a wandering domain.

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arXiv2020

Geometrically finite transcendental entire functions

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c for more restrictive classes of entire functions.

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arXiv2017

A landing theorem for entire functions with bounded post-singular sets

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called "periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called "filaments". We show that every periodic filament lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic filament. More generally, we prove that every point of a hyperbolic set is the landing point of a filament.

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arXiv2016

Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture

A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence a good understanding of these functions has implications in wider generality. Our goal is to study the topological properties of the Julia sets of entire functions of disjoint type. In particular, we give a detailed description of the topology of their connected components. More precisely, consider a "Julia continuum" C of such a function, i.e. the closure in the Riemann sphere of a component of the Julia set. We show that infinity is a terminal point of C, and that C has span zero in the sense of Lelek; under a mild geometric assumption on the function C is arc-like. (Whether every span zero continuum is also arc-like was a famous question in continuum theory, only recently resolved in the negative.) Conversely, we construct a single disjoint-type entire function with the remarkable property that each arc-like continuum with at least one terminal point is realised as a Julia continuum. The class of arc-like continua with terminal points is uncountable. It includes, in particular, the sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc, so these can all occur as Julia continua of a disjoint-type entire function. We give similar descriptions of the possible topology of Julia continua that contain periodic points or points with bounded orbits, and answer a question of Bara\'nski and Karpi\'nska by showing that Julia continua need not contain points that are accessible from the Fatou set. Furthermore, we construct an entire function whose Julia set has connected components on which the iterates tend to infinity pointwise, but not uniformly. This is related to a famous conjecture of Eremenko concerning escaping sets of entire functions.

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