Schedule

——— Tentative schedule ———

10.30-11.00 Arrival, tea and coffee

11.00-11.50 Mikhail Hlushchanka (University of Amsterdam)

12.00-12.25 Anna Jové (University of Barcelona)

12.30-13.55 Lunch

13.55-14.00 Discussion of next year's meeting

14.00-14.50 Gwyneth Stallard (The Open University)

14.55-15.45 David Martí-Pete (University of Liverpool)

15.45-16.15 Coffee break

16.15-16.40 Isabella Lewis (The Open University)

16.45-17.35 Weiwei Cui (Shandong University)

18.00 Dinner

——— Dinner ———

The dinner is planned to take place at the vegetarian restaurant Namaste Holborn, 33 Boswell Street, London WC1N 3BP.

Registration Form

Registration Form for Remote Participation

——— Abstracts ———

• Gwyneth Stallard (The Open University): 

Boundary dynamics for non-autonomous systems of inner functions

Abstract: We consider forward compositions of inner functions. These are holomorphic self maps of the unit disc that arise naturally, for example, when studying wandering domains in complex dynamics and, by taking radial limits we obtain maps from the boundary to itself that are defined almost everywhere. We discuss the dynamics for sequences of such boundary maps giving results related to shrinking targets, mixing and entropy that generalise results of Pommerenke. This is joint work with Anna Benini, Vasso Evdoridou, Nuria Fagella and Phil Rippon.

• Anna Jové (University of Barcelona): 

Lyapunov exponents for a family of tangent-like maps

I will discuss the Lyapunov exponent associated with a model family of meromorphic (tangent-like) maps with respect to harmonic measure. I will address its integrability, positivity, and its dependence on parameters within the family. Some of the methods and results extend beyond this setting and apply more generally to simply connected attracting basins of meromorphic maps. This is joint work in progress with Matthieu Astorg.

• David Martí-Peter (University of Liverpool): 

Wandering dynamics of transcendental functions

In the recent years there have been a lot of new constructions of transcendental functions with wandering domains. Together with Lasse Rempe and James Waterman, we proved that every full compact set is a wandering compact sets for some transcendental entire function. On the other hand, Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, Phil Rippon and Gwyneth Stallard classified the internal dynamics of wandering domains and also constructed several examples of each type. In my talk I will give an overview of these constructions and discuss a new and more general way of using approximation theory to obtain transcendental functions with wandering continua which allows us to prescribe the internal dynamics; more precisely, the resulting function is conjugate on the closure of a wandering domain to the model map used in the approximation. This allows us to obtain examples of simply connected wandering domains with any type of internal dynamics and different boundary dynamics. This is a joint work with Vasiliki Evdoridou and Lasse Rempe.

• Mikhail Hlushchanka (University of Amsterdam):

Canonical decomposition of rational maps

There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics, with celebrated examples due to William Thurston. The main goal of this talk is to present a powerful decomposition of rational maps based on the topological structure of their Julia sets, obtained jointly with Dzmitry Dudko and Dierk Schleicher: every postcritically finite rational map with non-empty Fatou set can be canonically decomposed into crochet maps, whose Julia sets are "thinly connected", and Sierpiński maps, whose Julia sets are "heavily connected". More precisely, crochet maps are those whose Julia sets are totally disconnected by the removal of a countable set of points, while Sierpiński maps are those whose Fatou components are Jordan disks with pairwise disjoint closures. At the end of the talk, I will discuss connections and applications of this result in conformal geometry and geometric group theory.

• Isabella Lewis (The Open University):

A Unified Approach to Ptolemy, Casey and Penner's Theorems

Ptolemy's Theorem asserts that for four points lying on a circle in the Euclidean plane, the product of the diagonals equals the sum of the product of the opposite side. Casey's Theorem is a generalisation of Ptolemy's Theorem to circles rather than points. Both theorems have generalisations to spherical and hyperbolic geometry and progress has been made in n-dimensional Euclidean space. Penner's Theorem, arising in decorated Teichmüller theory, is a similar theorem for horocycles. In this talk we use Lorentzian geometry to reinterpret these theorems and provide n-dimensional versions for all three geometries.

• Weiwei Cui (Shandong University):

Regularity of hairs of entire functions

Each component of Julia sets of disjoint type is called a Julia component. The topology of Julia components has been studied in details by Lasse Rempe. For the metric side, earlier works of Fatou, Karpinska and Baranski show that Julia components are often hairs with its Hausdorff dimension being equal one. In the joint work with David Marti-Pete, Leticia Pardo-Simon and Lasse Rempe, we construct disjoint type entire functions whose hairs have Hausdorff dimension and packing dimension being any prescribed number in [1,2], subject to trivial relations.