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View article: The Escaping Set in Transcendental Dynamics
The Escaping Set in Transcendental Dynamics Open
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 expla…
View article: Entire functions with Cantor bouquet Julia sets
Entire functions with Cantor bouquet Julia sets Open
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 arc…
View article: The escaping set in transcendental dynamics
The escaping set in transcendental dynamics Open
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 expla…
View article: Spiders’ Webs in the Eremenko–Lyubich Class
Spiders’ Webs in the Eremenko–Lyubich Class Open
Consider the entire function $f(z)=\cosh (z)$. We show that the escaping set $I\left(\,f\right)$—that is, the set of points whose orbits tend to infinity under iteration of $f$—has a structure known as a “spider’s web”. This disproves a co…
View article: Spiders' webs in the Eremenko-Lyubich class
Spiders' webs in the Eremenko-Lyubich class Open
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 conjectur…
View article: Bounded Fatou and Julia components of meromorphic functions
Bounded Fatou and Julia components of meromorphic functions Open
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 …
View article: Points of convergence -- music meets mathematics
Points of convergence -- music meets mathematics Open
"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…
View article: Entire functions with Cantor bouquet Julia sets
Entire functions with Cantor bouquet Julia sets Open
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 …
View article: The Eremenko–Lyubich constant
The Eremenko–Lyubich constant Open
Eremenko and Lyubich proved that an entire function whose set of singular\nvalues is bounded is expanding at points where its image has large modulus.\nThese expansion properties have been at the centre of the subsequent study of\nthis cla…
View article: Geometrically finite transcendental entire functions
Geometrically finite transcendental entire functions Open
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 quot…
View article: Bounded Fatou and Julia components of meromorphic functions
Bounded Fatou and Julia components of meromorphic functions Open
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 …
View article: Second order linear differential equations with a basis of solutions having only real zeros
Second order linear differential equations with a basis of solutions having only real zeros Open
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 o…
View article: Eremenko's conjecture, wandering Lakes of Wada, and maverick points
Eremenko's conjecture, wandering Lakes of Wada, and maverick points Open
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 conje…
View article: A bouquet of pseudo-arcs
A bouquet of pseudo-arcs Open
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 a…
View article: The Eremenko-Lyubich constant
The Eremenko-Lyubich constant Open
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 …
View article: Non-compact Riemann surfaces are equilaterally triangulable
Non-compact Riemann surfaces are equilaterally triangulable Open
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 e…
View article: Escaping sets are not sigma-compact
Escaping sets are not sigma-compact Open
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.
View article: Singular orbits and Baker domains
Singular orbits and Baker domains Open
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…
View article: Wandering lakes of Wada
Wandering lakes of Wada Open
We construct a transcendental entire function for which infinitely many Fatou components share the same boundary. This solves the long-standing open problem whether Lakes of Wada continua can arise in complex dynamics, and answers the anal…
View article: Geometrically finite transcendental entire functions
Geometrically finite transcendental entire functions Open
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 quot…
View article: Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials
Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials Open
Let $ P \colon \mathbb{C} \to \mathbb{C} $ be an entire function. A Poincaré function $ L \colon \mathbb{C} \to \mathbb{C} $ of $ P $ is the entire extension of a linearising coordinate near a repelling fixed point of $ P $. We propose suc…
View article: Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture Open
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…
View article: Arc-like continua, Julia sets of entire functions, and Eremenko's\n Conjecture
Arc-like continua, Julia sets of entire functions, and Eremenko's\n Conjecture Open
A hyperbolic transcendental entire function with connected Fatou set is said\nto be "of disjoint type". It is known that a disjoint-type function provides a\nmodel for the dynamics near infinity of all maps in the same parameter space;\nhe…