# Dynamical Systems

## Seminars in 2015-16

In 2015-16, the organiser of the Dynamical Systems seminar was **Dr Alexandre De Zotti**.

### Seminars

Speaker: **Nikita Sidorov** (University of Manchester)

Title: **Self-affine sets: topology and arithmetic**

Time: Wednesday May 18, 2016, 15:00-16:00

Place: Room 106

Abstract: Let M be an n x n real matrix with eigenvalues less than 1 in modulus. Consider the iterated function system (IFS) {Mx-v, Mx+v} with some vector v such that it is non-degenerate. In my talk I will address the questions related to the topology of the attractor of this IFS (connectedness, non-empty interior) as well as connections to beta-expansions and similar number-theoretic objects for n=2. This talk is based on my three recent papers with Kevin Hare (Waterloo).

Speaker: **David Simmons** (University of York)

Title: **Dimension gaps in self-affine sponges**

Time: Wednesday May 11, 2016, 15:00-16:00

Place: Room MATH-106

Abstract: In this talk, I will discuss a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension, as well as my recent result showing that the answer is negative. The counterexample is a self-affine sponge in $\mathbb R^3$ coming from anaffine iterated function system whose coordinate subspace projections satisfy the strong separation condition. Its dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, which implies that sponges with a dimension gap represent a nonempty open subset of the parameter space. This work is joint with Tushar Das (Wisconsin - La Crosse).

Speaker: **Olga Valba** (National Research University, Moscow)

Title: **Phase transitions in random networks**

Time: Tuesday April 19, 2016, 16:00-17:00

Place: Room MATH-106

Abstract: We consider an equilibrium ensemble of large Erdős-Renyi topological random networks with fixed vertex degree and two types of vertices, black and white, prepared randomly with the bond connection probability p. The network energy is a sum of all unicolor triples (either black or white), weighted with chemical potential of triples μ. Minimizing the system energy, we see for some positive μ the formation of two predominantly unicolor clusters, linked by a string of Nbw black-white bonds. We have demonstrated that the system exhibits critical behavior manifested in the emergence of a wide plateau on the Nbw(μ) curve, which is relevant to a spinodal decomposition in first-order phase transitions.

Speaker: **Vasso Anagnostopoulou** (Imperial College, London)

Title: **Sturmian measures in ergodic optimization**

Time: Wednesday April 13, 2016, 15:00-16:00

Place: Room MATH-106

Speaker: **Vasiliki Evdoridou** (Open University)

Title: **Fatou’s web and non-escaping endpoints**

Time: Wednesday March 9, 2016, 15:00-16:00

Place: Room MATH-106

Abstract: Let f be Fatou’s function, that is, f(z)= z+1+ exp(-z). We show that the escaping set of f, which consists of all points that tend to infinity under iteration, has a structure known as a spider’s web. We discuss a consequence of this result concerning the non-escaping endpoints of the Julia set of f. More specifically, we prove that the set of non-escaping endpoints together with infinity form a totally disconnected set. Finally, we show how these techniques can be adapted in order to show that a similar result holds for some functions in the exponential family.

Speaker: **Arnaud Chéritat** (Toulouse)

Title: **Straightening the square**

Time: Wednesday February 24, 2016, 15:00-16:00

Place: Room MATH-106

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

Title: **Landing of rays and dreadlocks for transcendental entire functions**

Time: Wednesday February 17, 2016, 15:00-16:00

Place: Room MATH-106

Abstract: There is a famous theorem, which we shall call the "Douady-Hubbard landing theorem", which states that every repelling or parabolic periodic point of a polynomial with bounded postcritical set can be accessed by a certain periodic curve (an “external ray”); conversely every such ray lands at a repelling or parabolic point. (I will give a short introduction to these concepts and the result in my talk.) The theorem has been a cornerstone of polynomial dynamics, being central to the famous “Yoccoz puzzle” which has been used to study the local connectivity of the Mandelbrot set, among other things.

We prove a version of this result for transcendental entire functions with bounded postsingular sets. In the case where the function has finite order, our result implies again that every repelling (or parabolic) periodic point is the landing point of a certain periodic “hair”. However, in the right formulation our result applies even to functions where the Julia set contains no nontrivial curves, where the role of the hairs is taken on by objects we call “dreadlocks”. Furthermore, we more generally establish landing of hairs or dreadlocks at points in hyperbolic subsets of the Julia set. (This is joint work with Anna Miriam Benini.)

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

Title: **Fractal dimensions of an overlapping generalization of Barański Carpets**

Time: Wednesday February 10, 2016, 15:00-16:00

Place: Room MATH-106

Abstract: We will study the Hausdorff, packing and box dimension of a family of self-affine sets generalizing Barański carpets. More specifically we fix a Barański system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction lets us give formulas for the fractal dimensions outside of a small set of exceptional translations. These formulae will coincide with those for the non-overlapping case, and thus provide us with examples where the box-counting and Hausdorff dimension do not necessarily agree. (This is joint work with Thomas Jordan).

Speaker: **Simon Albrecht** (Stony Brook University)

Title: **On functions in the Speiser class with one tract**

Time: Thursday November 12, 2015

Abstract: The Speiser class consists of all transcendental entire functions with finite singular set. A tract of an entire function f is a connected component of {z : |f(z)|>R} for some R>0. We construct functions in the Speiser class with a prescribed tract by using quasiconformal folding, a method introduced by C. Bishop in 2011.