# Dynamical Systems

## Seminars in 2016-17

In 2016-17, the organiser of the Dynamical Systems seminar was **Dr Simon Albrecht**.

### Seminars

Speaker: **Daniel Meyer** (Jacobs University)

Title: **Expanding Thurston maps**

Time: Thursday January 19, 2017, 13:00-14:00

Place: Room MATH-211

Abstract: In this talk I will give an overview of my (recent and not so recent) work, done jointly with Mario Bonk. An ``expanding Thurston map'' is a postcritically finite branched covering map of the 2-sphere that is expanding in a suitable sense. We study these maps with methods from geometric group theory. A major motivation comes from ``Cannon's conjecture''. This well-known conjecture stipulates that a group that ``behaves topologically'' as a Kleinian group ``is'' a Kleinian group. This conjecture corresponds to Thurston's characterization of rational maps among Thurston maps. Closely related is the problem when a metric sphere is quasisymmetrically equivalent to the standard unit sphere, i.e., a ``quasisphere''. We define ``visual metrics'' for an expanding Thurston map f. The geometry of the sphere S^2 equipped with such a metric d encodes properties of f. For example, (S^2,d) is a quasisphere if and only if f is topologically conjugate to a rational map.

Speaker: **Adam Epstein** (University of Warwick)

Title: **Transversality**

Time: Tuesday December 6, 2016, 15:00-16:00

Place: MAGIC Room G-16

Abstract: Thurston's infinitesimal rigidity argument, and extensions due to the speaker and various other researchers, yields transversality principles via the Inverse Function Theorem. It has been suggested that our own account constitutes folklore more appropriate to a conference proceedings than to a research journal. We compare our own results to those subsequently claimed by others, and highlight differences in applicability and reliability.

Speaker: **Kirill Lazebnik** (Stony Brook University)

Title:** Several constructions in the Eremenko-Lyubich class**

Time: Tuesday November 29, 2016, 15:00-16:00

Place: MAGIC Room G-16

Abstract: In 1985, Sullivan proved the no-wandering domain theorem for rational maps of the Riemann Sphere. Using the same ideas, (Lyubich and Eremenko) and (Goldberg and Keen), proved a no-wandering domain theorem for entire functions with finite singular set. Whereas wandering domains were known to exist for more general entire functions, it was unknown whether wandering domains occur for entire functions with bounded (but possibly infinite) singular set. Such a function was constructed by Bishop in 2012 using the "folding theorem" (proven in the same paper) for constructing entire functions. We will first discuss the folding theorem and the wandering domain example. Next we will discuss the topology of the wandering domain. Finally we will show how these methods can be used to construct several functions in the Eremenko-Lyubich class with other dynamical properties we will prescribe.

Speaker: **Paul Verschueren** (Open University)

Title: **Quasiperiodic sums and products**

Time: Tuesday November 22, 2016, 15:00-16:00

Place: MAGIC Room G-16

Abstract: Quasiperiodic Sums (Birkhoff Sums over a rotation) and Products arise in many areas of mathematics including the study of Strange Non-Chaotic Attractors, Critical KAM Theory, Quantum Chaos, q-series, Partition Theory, and Diophantine Approximation.

Graphs of these functions can form intriguing geometrically strange and self-similar structures. They are easy and rewarding to investigate numerically, and suggest many avenues for investigation. However they prove resistant to rigorous analysis.

In this talk we will survey some of the most important examples, and focus on the most heavily studied example, Sudler's product of sines. We will also report on new approaches which allow us to settle negatively an open question of Erdős & Szekeres from 1959, and to prove a number of experimental results reported recently by Knill & Tangerman (2011).

Speaker: **Fabrizio Bianchi** (Imperial College, London)

Title: **Holomorphic motions of Julia sets: a lambda-lemma in several complex variables**

Time: Tuesday November 15, 2016, 15:00-16:00

Place: Room G16 (MAGIC)

Abstract: For a family of rational maps, results by Lyubich, Mané-Sad-Sullivan and DeMarco provide a fairly complete understanding of dynamical stability. A central tool in this description is the celebrated lambda-lemma, which allows one to promote a holomorphic motion of a set to one of its closure.

Starting from the basics, and paying particular attention to the differences with respect to the transcendental case, I will review the one-dimensional theory and depict a general panorama about dynamical stability in several variables. I will focus on the arguments that do not readily generalise to this setting, and introduce the tools and ideas that allow one to overcome these problems.

Speaker: **David Sixsmith** (University of Liverpool)

Title: **On class A and class B **

Time: Tuesday November 8, 2016, 15:00-16:00

Place: Room G16 (MAGIC)

Abstract: In 1970 MacLane asked if it is possible for a locally univalent function in the class A to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class A.

Speaker: **David Martí-Pete** (Open University)

Title:** The escaping set of transcendental self-maps of the punctured plane**

Time: Tuesday November 1, 2016, 15:00-16:00

Place: Room G16 (MAGIC)

Abstract: We study the iteration of holomorphic self-maps of C*, the complex plane with the origin removed, for which both zero and infinity are essential singularities. The escaping set of such maps consists of the points whose orbit accumulates to zero and/or infinity following what we call essential itineraries. We show that the Julia set always contains escaping points with every essential itinerary. The concept of essential itinerary leads to a partition of the escaping set into uncountably many disjoint sets, the boundary of each of which is the Julia set. Under certain hypotheses, each of these sets contains uncountably many curves to zero and infinity. We also use approximation theory to provide examples of functions with escaping Fatou components.

Speaker: **Simon Albrecht** (University of Liverpool)

Title: **Speiser class Julia sets with dimension near 1**

Time: Tuesday October 25, 2016, 15:00-16:00

Place: Room G-16 (MAGIC)

Abstract: We prove that for any d>0, there exists a transcendental entire function f with only finitely many singular values such the Hausdorff dimension of the Julia set of f is less than 1+d. This is joint work with Christopher J. Bishop from Stony Brook.

Speaker: **André Salles de Carvalho** (University of São Paulo, Brazil)

Title: **Inverse limits and measurable pseudo-Anosov maps**

Time: Tuesday October 18, 2016, 15:00-16:00

Place: Room G16 (MAGIC)

Abstract: I'll present a construction which starts with a graph endomorphism and, using a quotient of its inverse limit, yields:

- a pseudo-Anosov map if the graph map is a train track map;
- a generalized pseudo-Anosov map if the graph map is post-critically finite and has an irreducible aperiodic transition matrix;
- an interesting type of surface homeomorphisms which generalizes both the previous classes otherwise.

In particular, this produces a unified construction of surface homeomorphisms whose dynamics mimics that of the tent family of interval endomorphisms, completing an earlier construction of unimodal generalized pseudo-Anosov maps in the post-critically finite case. This is joint work with Phil Boyland and Toby Hall.

Speaker: **Patrick Comdühr** (Kiel University)

Title: **Quasiregular maps and Zorich maps with one attracting fixed point**

Time: Tuesday September 6, 2016, 10:45-11:45

Place: Room MATH-117

Abstract: The theory of holomorphic functions in one complex variable is well known by students and is of rich structure. We want to focus on functions in R^d, with d > 2, which behave in a similar way as their holomorphic counterparts. More precisely we look at a quasiregular analogue of the exponential map and use iteration theory to understand its behaviour.