2025

This year's speakers will be Prof. Mario Bonk (UCLA), Dr Vasiliki Evdoridou (Open University), Josef Greilhuber (Stanford University), Jocelyn Russell (Imperial London), Prof. Caroline Series (University of Warwick) and Dr Nina Snigireva (University of Galway).

The meeting will take place at De Morgan House in London.

The One Day Function Theory Meeting 2025 is organised by Daniel Meyer and Julia Münch and funded by the London Mathematical Society.

——— Schedule ———

——— Abstracts ———

• Mario Bonk (UCLA): 

Thurston’s pull-back map, invariant covers, and the global dynamics on curves

A postcritically-finite rational map f on the Riemann sphere induces a holomorphic map on an associated Teichmüller space, the Thurston pull-back map σf . The boundary behavior of σf is closely related to dynamical properties of simple closed curves under the pull-back operation by f . A stubborn open problem in this area is the existence of a “finite global curve attractor” for this operation. For maps f with four postcritical points this problem can be phrased in terms of the boundary behavior of holomorphic maps on the upper half-plane that are “polymorphic” with respect to the modular group. In my talk I will give an introduction to this topic and report on some recent joint work with M. Hlushchanka and R. Lodge.

• Josef Greilhuber (Stanford University): 

Hypersurfaces on which few harmonic functions vanish

It is easy to see that if one harmonic function vanishes on a given set in the Euclidean plane, then infinitely many linearly independent harmonic functions do. Perhaps surprisingly, this is no longer true in higher dimensions. We will show that in any dimension greater than two, there exist cones on which exactly two linearly independent harmonic functions vanish. This result holds on the level of germs at the origin. We will also show that smooth hypersurfaces can exhibit the same phenomenon if one asks for harmonic functions defined on a large enough (but still bounded) domain in Euclidean space. This last observation rests on a surprising unique continuation result in ellipsoidal coordinates.

• Nina Snigireva (University of Galway):

Weakly Contractive and Noncontractive Iterated Function Systems

We  will begin by discussing weakly contractive iterated function systems (IFSs). We will then consider noncontractive IFSs and investigate techniques which prove useful in studying such IFSs. In particular, we will concentrate on the question of the existence of semiattractors and attractors for noncontractive IFSs. We will construct explicit examples of noncontractive IFSs which admit semiattractors/attractors and discuss the tools which can be used to build such examples. 

This is joint work with K. Lesniak, F. Strobin and A. Vince.

• Vasiliki Evdoridou (Open University): 

Wandering dynamics of transcendental functions 

In recent years there have been a lot of new constructions of transcendental functions with wandering domains. Marti-Pete,  Rempe and Waterman proved that every full compact set is a wandering compact set for some transcendental entire function. Moreover, in a joint work with Benini, Fagella, Rippon and Stallard we classified the internal dynamics of wandering domains and also constructed  examples of each type. In this talk we will 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 Jordan wandering domains with any type of internal dynamics and different boundary dynamics. 

This is joint work with David Marti-Pete and Lasse Rempe.

• Jocelyn Russell (Imperial London):

Toy Model for Bi-Critical Irrationally Indifferent Attractors

Holomorphic maps which have fixed points with multiplier equal to an irrational rotation exhibit complicated and rich long term behaviour. The development of the Inou-Shishikura class led authors such as Cheraghi to study critical orbits via a special non-holomorphic toy model which retains important topological and geometrical properties. In this talk I will introduce some new ideas about the bi-critical version of the toy model, for which the extra critical point induces much more complexity.

• Caroline Series (University of Warwick):

Exploring the tree of primitive traces for two generator subgroups of SL(2,C)

It is well known that, with minor exceptions, a two generator subgroup of SL(2,C) is determined up to conjugation by the triple of traces (Tr A, Tr B, Tr AB). 

Conversely, any triple of complex numbers defines a representation a->A, b->B of the free group F on two generators (a,b) into SL(2,C).

 It is natural to ask about the properties of G = <A,B>, the most obvious being whether  or not it is discrete.  The Nielsen moves which replace generators (a,b) by (a,ab) or (b,ab) allow one to run through all possible generating pairs and hence through all possible trace triples. 

Moreover these can be conveniently arranged around the vertices of a trivalent tree. Say this tree of traces is totally divergent if there are only finitely many traces below any given bound.

Bowditch's condition and primitive stability are two apparently quite different conditions on the images of primitive elements (generating elements of F) in G under which the trace tree is totally divergent. They have recently been shown to be equivalent and together show that under certain circumstances the trace tree is totally divergent even when G is not discrete. Moreover a small relaxation of Bowditch's condition is equivalent to total divergence.

We illustrate with various pictures in complex parameter spaces which demonstrate these results and suggest some conjectures and unanswered questions.