# Dynamical Systems

## Our work

At the University of Liverpool, there is a large and active research group in dynamical systems:

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.

Dr Radhakrishnan Nair works on ergodic theorems and their intimate relation to aritmetic questions. Ergodic theorems are foundational to ergodic theory (also called measurable dynamics). Their ambit also covers a number of topics perhaps not self evidently related to ergodic theory. A number of famous theorems in mathematics are in fact ergodic theorems. We mention a few. Dirichlet's theorem on primes in arithmetic progressions, Fejer's theorem on the summability of Fourier series, Lebesgue's differentiation theorem, the law of large numbers -- both weak and strong, Jarnik's theorem on the set of badly approximable numbers, Borel's theorem on normal numbers, Szemeredi's theorem on arithmetic progressions in sets of integers of positive density and so on. This is only a small sample of the interactions of the subject with other branches of mathematics, but even this list hints at its broad impact and also the recurrence of arithmetic topics throughout its development. The particular lines of enquiry that have most interested Dr. Nair include the following:

- A: subsequence ergodic theorems, their proofs and applications;
- B: distribution modulo one and its relation to ergodic theorems;
- C: the failure of ergodic theorems and its relation to diophantine approximation and hyperbolic dynamics;
- D: Glasner sets, toplogical dynamics and exponential sums;
- E: the existance of invariant measures for maps of the interval;
- F: non-linear Poincaré and multiple recurrence and the arithmetic of sets of integers of positive density;
- G: strong uniform distribution;
- H: Riemann sums and Lebesgue integrals;
- I: exceptional sets in hyperbolic and self-affine settings;
- J: metric discrepancy estimates;
- K: ergodic square functions and oscillation functions;
- L: moving average ergodic theorems and their applications.

Papers related to these sections are indicated on Dr Nair's homepage by an appropriate letter, though these classifications are only a rough guide.

Prof Lasse Rempe works in Complex Dynamics. He is particularly interested in the dynamics of transcendental functions of one complex variable, e.g. transcendental entire functions such as the exponential map, or transcendental meromorphic functions such as the tangent map. His doctoral thesis, under the supervision of Walter Bergweiler in Kiel, obtained a number of new results on the family of exponential maps. In particular, he was able to answer questions posed by Herman, Baker and Rippon and by Devaney, Goldberg and Hubbard in the 1980s.

Since then, his main interest has been to understand the dynamics of larger classes of entire functions, particularly with a view towards their topology. Joint work with Rottenfußer, Rückert and Schleicher answered long-standing open questions by Fatou and Eremenko, and there has been significant progress in studying the dynamics of the large Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values.

Current projects include a question of Eremenko regarding unboundedness of escaping components, the study of measurable dynamics of transcendental functions following ideas of Urbanski, Zdunik and others, and (jointly with Sebastian van Strien) density of hyperbolicity in parameter spaces of real transcendental entire functions.