David Martí-Pete

• PhD Mathematics, The Open University
• Lecturer at University of Liverpool

We study the escaping set of functions in the class B*, that is, holomorphic functions f : C* → C* for which both zero and infinity are essential singularities, and the set of singular values of f is contained in an annulus. For functions in the class B*, escaping points lie in their Julia set. If f is a composition of finite order transcendental s...

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 b...

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 functi...

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 functi...

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 analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. Our theorem also provi...

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...

Recently, Bishop constructed the first example of a bounded‐type transcendental entire function with a wandering domain using a new technique called quasiconformal 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 class B with wandering...

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 Bake...

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 the...