2024

The One Day Function Theory Meeting 2024 took place on Monday, 2 September 2024 at the Department of Mathematics of University College London. The meeting was organised by Leticia Pardo-Simón (Universitat de Barcelona) and David Martí-Pete (University of Liverpool). The organisers are very grateful to the Heilbronn Institute for funding this meeting, and to Rod Halburd for hosting the meeting at UCL.

——— Schedule ———

——— Abstracts ———

• Luka Boc Thaler (University of Ljubljana)
  On the exponential tower functions 

Abstract: We introduce a new class of entire functions $\mathcal{E}$ (exponential tower functions) and discuss its properties. In particular we will prove that $\mathcal{E}$ is a dense subset of the set of all non-vanishing entire functions $\mathcal{O}^*(\mahtbb{C})$, and that for every closed set $V\subseteq \mathbb{C}$ that contains the origin and at least one more point, there exist a locally univalent function in our class $\mathcal{E}$  whose set of singular values is equal to $V$.

• Julia Münch (University of Liverpool)
  Extending rational expanding Thurston maps

Abstract: Suppose we are given a holomorphic map on the Riemann sphere, can we extend it to a map on 3-dimensional Euclidean space? Of course we would like to extend it to a map that has good regularity properties.Since there will not be a holomorphic map, we will work towards obtaining a slightly weaker property, namely a uniformly quasi-regular map. This has been done for example for Lattès maps - those are a subfamily of expanding rational Thurston maps. We are aiming to extend this result to the whole class by a different method.

• Julia Slipantschuk (University of Warwick)
  Resonances for hyperbolic maps

Abstract: I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.

• Leon Staresinic (Imperial  College London)
  Density of Stable Interval Translation Maps 

Abstract: Interval Translations Maps (ITM’s) are a natural generalisation of the well-known Interval Exchange Transformations (IET’s). They are obtained by dropping the bijectivity assumption for IET’s. There are two basic types of ITM’s, finite-type and infinite-type ones. They are classified by their non-wandering sets: it is a finite union of intervals for finite-type maps and a Cantor set for infinite-type maps. One of the basic questions in the field is: How prevalent is each type of map in the parameter space? In this work, we show that the set of stable finite-type maps forms an open and dense subset in the parameter space of ITM’s with a fixed number of intervals. This is joint work with Sebastian van Strien and Kostya Drach.

• Alan Sokal (UCL)
  Some conjectures concerning the zeros of the deformed exponential function 

Abstract: I discuss some interesting conjectures concerning the zeros of the deformed exponential function $F(x,y) = \sum\limits_{n=0}^\infty {x^n \over n!} y^{n(n-1)/2}$. Some of these are related to more general conjectures concerning the coefficientwise nonnegativity of the Taylor expansion for the leading root of certain series $f(x,y) = \sum\limits_{n=0}^\infty \alpha_n x^n  y^{n(n-1)/2}$.