# Dynamical Systems

## Seminars in 2017-18

In 2017-18, the organiser of the Dynamical Systems seminar was **Dr Simon Albrecht**.

### Seminars

Speaker: **Leticia Pardo Simón** (University of Liverpool)

Title: **Cantor Bouquets and escaping singular orbits in Class B**

Time: Thursday April 26, 2018, 13:00-14:00

Place: Room MATH-514

Abstract: As a partial answer to Eremenko's conjecture, it is known for functions in Class B of finite order that every point in their escaping set can be connected to infinity by an escaping curve. Even if those curves, called "hairs" or "rays" not always land, this has been positively proved for some functions with bounded postsingular set by showing that their Julia set is structured as a Cantor Bouquet, that is, an embedding in the plane of a straight brush. In this talk I will consider certain functions with unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, some hairs will split when they hit critical points. Still, I will conclude that the existence of a map on their parameter space whose Julia set is a Cantor Bouquet guarantees that such hairs, if maybe now with split ends, still land.

Speaker: **Tuomas Sahlsten** (University of Manchester)

Title: **Entropy conjecture in uniformly quasiregular dynamics**

Time: Thursday April 19, 2018, 13:00-14:00

Place: Room MATH-514

Abstract: We establish Shub's entropy conjecture for all uniformly quasiregular mappings on compact connected oriented Riemannian manifolds without boundary such that their rational cohomology is not isomorphic to that of the sphere. The proof builds on the recent absolute continuity theorem of Kangasniemi for the equilibrium measure Okuyama and Pankka constructed for uniformly quasiregular maps and cohomology results by Kangasniemi and Pankka for manifolds supporting uniformly quasiregular dynamics.

This is a joint work with Ilmari Kangasniemi (Helsinki), Yusuke Okuyama (Kyoto) and Pekka Pankka (Helsinki)

Speaker: **Stephen Worsley** (University of Liverpool)

Title: **Arclike and indecomposable dynamic rays**

Time: Thursday April 12, 2018, 13:00-14:00

Place: Room MATH-514

Abstract: The dynamic rays of a function give structure its dynamics. The topology of these rays may not, however, be straightforward even for simple maps. I have studied an example of the transcendental entire function E(z)=2\pi i e^z, for which it is known that the closures of such rays may be an indecomposable continuum, a particularly pathological kind of continuum. I have shown that, furthermore, that the closure (in an appropriate compactification of the complex plane) of any such ray of E must be Arclike and I have given conditions on precisely which rays have indecomposable closure.

Speaker: **Keivan Mallahi-Karai** (Jacobs University)

Title: **Homogeneous dynamics and S-arithmetic quantitative Oppenheim conjecture**

Time: Thursday March 1, 2018, 13:00-14:00

Place: Room MATH-514

Abstract: Let $q$ be a non-degenerate indefinite quadratic form in $n>2$ variables over $ \mathbb{R}$, which is not a multiple of a rational form. Answering a longstanding question of Oppenheim, Margulis proved in 1986 that the set of values $q( \mathbb{Z}^n)$ is a dense subset of $ \mathbb{R}$. Quantifying this result, Eskin, Margulis, and Mozes obtained the asymptotic behavior of the number of integral vectors $v$ of norm at most $T$ satisfying $q(v) \in (a,b)$. Both works are dynamical in nature, and rely heavily on features of the unipotent flow dynamics on homogenous spaces.

In this talk, I will elaborate on this history and then discuss a recent generalisation of the theorem of Eskin, Margulis, and Mozes, in which instead of one real quadratic form, a finite number of quadratic forms over different completions of the set of rational numbers (real and $p$-adic) is considered. This talk is based on a joint work with Seonhee Lim and Jiyoung Han.

Speaker: **Irene Pasquinelli** (University of Durham)

Title: **Cutting sequences on Veech surfaces**

Time: Thursday February 15, 2018, 13:00-14:00

Place: Room MATH-514

Abstract: Consider the dynamical system given by the geodesic flow on a flat surface. Given a polygonal representation for a surface, one can code the trajectory using the sides of the polygons and thus obtain a cutting sequence. A natural question to ask then, is whether any sequence one picks can come from a certain trajectory. In other words, can we characterise the set of cutting sequences in the set of all sequences in the alphabet? And when the answer is yes, can we recover the direction of the trajectory? In this talk we will give an overview of the cases where these questions have been answered.

Speaker: **Jonathan Fraser** (University of St. Andrews)

Title: **Geometrically finite Kleinian groups and dimension**

Time: Friday February 9, 2018, 11:00-12:00

Place: Room MATH-104

Abstract: Kleinian groups act discretely on hyperbolic space and give rise to beautiful and intricate mathematical objects, such as tilings and fractal limit sets. The dimension theory of these limit sets has a particularly interesting history, the first calculation of the Hausdorff dimension going back to seminal work of Patterson from the 1970s. In the geometrically finite case, the Hausdorff, box-counting, and packing dimensions are all given by the Poincare exponent. I will discuss recent work on the Assouad dimension, which is not necessarily given by the Poincare exponent in the presence of parabolic points.

Speaker: **Luke Warren** (University of Nottingham)

Title: **Slow escaping points for quasimeromorphic mappings**

Time: Thursday December 14, 2017, 11:00-12:00

Place: Room TP-117

Abstract: For a transcendental meromorphic function f, the escaping set is given by I(f) = {x in C : f^n(x) is defined for all n, |f^n(x)| tends to infinity as n tends to infinity}.

It has been shown by Rippon and Stallard that for such f, there exists a point in J(f) that escapes arbitrarily slowly. More recently, a result by Nicks states that the slow escaping result also holds for quasiregular mappings of transcendental type, which are higher dimensional analogues of analytic mappings with an essential singularity at infinity.

Following a similar method of Nicks, combined with some ideas on the escaping set of quasimeromorphic mappings with an infinite number of poles, we shall extend this result and show that there exists a point that escapes arbitrarily slowly for quasimeromorphic mappings with an essential singularity at infinity. This will include the proof of a new growth result for quasiregular mappings near an essential singularity.

Speaker: **Alexandre De Zotti** (Imperial College, London)

Title: **Hausdorff dimension of the boundary of Siegel disks**

Time: Thursday December 7, 2017, 11:00-12:00

Place: Room TP-117

Abstract: In this work I will present my work in progress with Davoud Cheraghi towards a proof of the existence of Siegel disks of quadratic polynomials with a boundary of Hausdorff dimension two.

Speaker: **Nikos Karaliolios** (Imperial College, London)

Title: **Cohomological Rigidity and the Anosov-Katok construction**

Time: Thursday November 30, 2017, 11:00-12:00

Place: Room TP-117

Abstract: Let f be a smooth volume preserving diffeomorphism of a compact manifold and \phi a known smooth function of zero integral with respect to the volume. The linear cohomological equation over f is \psi \circ f - \psi = \phi where the solution \psi is required to be smooth. Diffeomorphisms f for which a smooth solution \psi exists for every such smooth function \phi are called Cohomologically Rigid. Herman and Katok have conjectured that the only such examples up to conjugation are Diophantine rotations in tori.

We study the relation between the solvability of this equation and the fast approximation method of Anosov-Katok and prove that fast approximation cannot construct counter-examples to the conjecture.

Speaker: **Kai Rajala** (University of Jyväskylä)

Title: **Uniformization of metric surfaces**

Time: Monday November 27, 2017, 11:00-12:00

Place: Room TP-117

Abstract: We discuss extensions of the classical uniformization theorem to metric spaces that are topological surfaces and have locally finite two-dimensional Hausdorff measure.

Speaker: **Lasse Rempe** (University of Liverpool)

Title: **A landing theorem for entire functions with bounded postsingular sets**

Time: Thursday November 16, 2017, 11:00-12:00

Place: Room MATH-117

Abstract: Let f be a polynomial in one complex variable, of degree 2. Assume that all critical points of f have bounded orbits under iteration of f. Then the Julia set of f is connected, and its unbounded connected component (the basin of infinity) is stratified by so-called “external rays” to infinity, which are the gradient lines for the Green’s function. These rays have played a crucial role in the study of polynomial dynamics for more than three decades, including in celebrated results of Yoccoz, McMullen and Lyubich.

A theorem of Douady states that every repelling periodic point of f is the landing point of a periodic external ray. This theorem has been the cornerstone of the above-mentioned breakthroughs in polynomial dynamics. We establish an analogous result for transcendental entire functions.

Of course, here the Julia set is unbounded, and there is no longer a basin of infinity. It was proposed already thirty years ago that certain curves, called “hairs”, on which the iterates tend to infinity can play the same roles as external rays in this setting. However, we now know that there are entire functions for which there are no such hairs at all. Instead, we use a notion of “dreadlocks” – certain unbounded connected sets – to prove the following result:

Suppose that f is a transcendental entire function, and assume that all singular values of f have bounded orbits under the iteration of f. Then every repelling periodic point of f is the landing point of a periodic dreadlock.

My goal in this talk is to give only a short introduction to make sense of the results, and then explain a crucial element of the proof of the theorem. (Joint work with Anna Benini).

Speaker: **Mark Holland** (Exeter University)

Title: **On recurrence statistics and Poisson laws for dynamical systems**

Time: Thursday October 26, 2017, 11:00-12:00

Place: Room MATH-117

Abstract: For a time series of observations (X_n) generated by a measure preserving dynamical system, we consider the maxima process M_n=Max(X_1,...X_n), and examine the probabilistic limit laws that can arise for M_n under suitable time normalization. We consider the implication these laws have on the recurrence statistics for the dynamical system, such as: extreme and record statistics, Poisson laws, and Borel Cantelli results.