Dr Toby Hall works in Topological Dynamics, with particular emphasis on surface homeomorphisms. He has a long-standing interest in the mechanism of horseshoe creation. His recent work, in collaboration with Philip Boyland and Andre de Carvalho, has focussed on the construction and properties of families of sphere homeomorphisms arising from inverse limits of families of unimodal maps.
Dr David Martí-Pete works in Complex Dynamics. More precisely, he studies the iteration of transcendental entire (or meromorphic) functions. In his doctoral thesis, under the supervision of Prof Phil Rippon and Prof Gwyneth Stallard at The Open University, he studied the escaping set of transcendental self-maps of the punctured plane. An example of such maps is the complex Arnold standard family.
Currently he is working on the topology and dynamics of wandering domains (and their boundaries). In a recent work with Lasse Rempe and James Waterman, they proved that wandering domains of entire functions may form Lakes of Wada and that escaping wandering domains may have non-escaping points on their boundaries (known as maverick points). They also constructed a counterexample to Eremenko’s conjecture concerning the bounded components of the escaping set of transcendental entire functions.
He is also interested in the Hausdorff dimension and computational complexity of Julia sets and several subsets thereof.
Dr Daniel Meyer works in geometric function theory and dynamics. He is particularly interested in quasiconformal and quasisymmetric maps, Thurston maps, and mating of polynomials.
Quasiconformal and quasisymmetric maps are maps that generalize the notion of conformal maps. They appear in a variety of settings, including complex analysis and geometric group theory. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetrically equivalent to some standard space. Of particular interest is the question when a space is a quasisphere, i.e., the quasisymmetric image of the standard 2-sphere.
A Thurston map is a map that acts topologically as a (postcritically finite) rational map. Thurston gave an answer when such a map ``is'' a rational map in a suitable way. Such maps were studied by Daniel together with Mario Bonk in a systematic fashion, where usually an expansion property was assumed. For such maps a so-called visual metric can be defined. The sphere equipped with such a metric is a quasisphere if and only if the given map ``is'' a rational map.
Another research interest of Daniel concerns the mating of polynomials. This is an operation that combines the filled Julia sets of two polynomials to form a new dynamical system. Surprisingly, this often results in a rational map. Conversely, one may ask if a given rational map arises in this way. Indeed, in the setting of postcritically finite, expanding maps (i.e., expanding Thurston maps), the answer is always positive: each sufficiently high iterate of such a map is a mating.
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