2023

The One Day Function Theory Meeting 2023 will take place on Monday, 4 September 2023 at De Morgan House, London.

——— Schedule ———

Leverhulme Lecture: The  visit of Prof. Alexandre Eremenko to the UK is funded by a Leverhulme Visiting Professor grant, this being the first of his Leverhulme lectures.

——— Abstracts ———

• Dimitrios Betsakos (Aristotle University of Thessaloniki)
  Semigroups of holomorphic functions; properties of the orbits
  Abstract: A family of holomorphic functions $\phi_t: \mathbb D \to \mathbb D$, $t\geq 0$, is called a semigroup if 

(a) $\phi_0(z)=z$,

(b) $\phi_{s+t}=\phi_s\circ \phi_t$, 

(c) $\phi_t(z)$ is continuous in $t$ for every $z$ in the unit disk $\mathbb D$.

If $z\in \mathbb D$, the curve $\gamma_z: [0,\infty)\to \mathbb D$ with $\gamma_z(t)=\phi_t(z)$ is the orbit of the semigroup starting from $z$. We will present some geometric properties of the orbits and study their asymptotic behaviour as $t\to\infty$. The main tools for the proofs of these properties are harmonic measure and hyperbolic geometry.

• Andrew Brown (University of Liverpool)
  Constructing slow-growing counterexamples to the Strong Eremenko Conjecture
  Abstract: The strong Eremenko Conjecture was disproved in the paper of Rottenfußer, Rückert, Rempe and Schleicher with a counterexample function in the Eremenko–Lyubich Class B with infinite order growth. It was also shown in the same paper that the conjecture holds for functions of finite order growth (and finite compositions of them). This talk will discuss the construction of counterexample functions with infinite order growth that grow asymptotically to finite order functions. 
  
  

• Alexandre Eremenko (Purdue University)
  Real zeros of solutions of linear differential equations
  Abstract: We determine all possible orders of entire functions A with the property that the differential equation w''+Aw=0 has two linearly independent solutions whose all zeros are real. Moreover, we show such functions A are of completely regular growth in the sense of Levin and Pfluger.
  Based on the joint work with Walter Bergweiler and Lasse Rempe.
  
  

• Natalia Jurga (University of St Andrews)
  Hausdorff dimension of the Rauzy gasket
  Abstract: The Rauzy gasket, which is the attractor of an iterated function system on the projective plane, is an important subset of parameter space in numerous dynamical and topological problems. Arnoux conjectured that the Hausdorff dimension of the Rauzy gasket is strictly less than 2, and since then there has been considerable interest in computing its Hausdorff dimension.
  We will discuss how the Rauzy gasket is naturally related to limit sets of Mobius maps acting on the extended real line. After introducing the set and discussing its history, we will show how recent developments in the theory of self-affine sets and measures can be adapted to compute the Hausdorff dimension of the Rauzy gasket.
  
  

• Maria Kourou (University of Würzburg)
  Speeds of convergence for petals of one-parameter semigroups in the unit disc
  Abstact: We discuss the backward dynamics of one-parameter semigroups of holomorphic self-maps of the unit disc. For this reason, we restrict to certain subsets of the unit disc called petals.
  The speeds of convergence for petals of one-parameter semigroups are introduced. These are analogous to speeds of convergence for one-parameter semigroups introduced by Bracci, yet profoundly different due to the nature of backward dynamics. Moreover, the asymptotic behavior of speeds of convergence for petals is discussed in correlation with the type of the petal. The results presented are based on a joint work with K. Zarvalis.
  
  

• Matteo Lotriglia (University College Dublin)
  Quadratic Misiurewicz parameters in a neighbourhood of −2
  Abstract: The main idea of the talk is to show that quadratic Misiurewicz parameters are real in a neighbourhood of $-2$. More specifically, we prove that the orbit $\{f^n(c)\}_{n \geq 1}$ of the quadratic polynomials $f(z)=z^2+c$ diverges as $n \to \infty$, if we choose $c \in \mathbb{C} \setminus \mathbb{R}$ in a neighbourhood of $-2$ and, $\forall n \geq 1$, we keep the orbit outside a ball centered in $0$.