# Dynamical Systems

## Seminars in 2019-20

In 2019-20, the organisers of the Dynamical Systems seminar were Dr Daniel Meyer (until December 2019) and **Dr James Waterman** (from January 2020).

### Seminars

Speaker: **Krzysztof Barański** (Warsaw University)

Title: **On the dimension of sets of points escaping to infinity at given rate under exponential iteration**

Time: Thursday July 2, 2020, 13:00-14:00 local time

Place: online (via MS Teams)

Abstract: We determine the Hausdorff and packing dimension of sets of points which escape to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we answer a question of Sixsmith from 2016. This is a joint work with Bogusława Karpińska.

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

Title: **Constructing examples of oscillating wandering domains**

Time: Thursday June 18, 2020, 13:00-14:00

Place: online (via MS Teams)

Abstract: Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. The first wandering domain of oscillating type was constructed by Eremenko and Lyubich in 1987. Motivated by their construction and the recent classification of simply connected wandering domains obtained by Benini, E., Fagella, Rippon and Stallard, we give a general technique, based on Approximation Theory, for the construction of bounded oscillating wandering domains. We show that this technique can be used to produce examples of oscillating wandering domains of all six different types that arise by the classification. This is joint work with P. Rippon and G. Stallard.

Speaker: **Alastair Fletcher** (Northern Illinois University)

Title: **Cantor sets and Julia sets**

Time: Thursday June 11, 2020, 16:00-17:00

Place: online (via MS Teams)

Abstract: Cantor sets embedded in R^n can arise dynamically in various ways, for example as Julia sets of uniformly quasiregular (uqr) mappings, or as attractor sets of iterated function systems. The main theme of this talk will be to try and recognize when an embedded Cantor set is a Julia set of a uqr map, or look for geometric properties of the Cantor set that preclude this. This is based on joint work with Vyron Vellis (UTK).

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

Title: **Docile entire functions**

Time: Thursday May 7, 2020, 13:00-14:00

Place: online (via MS Teams)

Abstract: For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle.

For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of “docile” functions. We also show docility for a large class of transcendental entire functions with bounded postsingular sets (those that are “strongly geometrically finite”). This is joint work with Mashael Alhamed and Dave Sixsmith.

In this talk, I will focus on explaining the definition of docile functions, and its motivation.

Speaker:** James Waterman** (University of Liverpool)

Title: **Iteration in tracts**

Time: Thursday April 30, 2020, 13:00-14:00

Place: online (via MS Teams)

Abstract: By a result of Rippon and Stallard, for a transcendental entire function there exist points in the escaping set whose orbit escapes to infinity arbitrarily slowly. We will discuss extending this result to prove the existence of points which escape to infinity arbitrarily slowly within a particular type of domain, called a tract.

Speaker: **Mary Rees** (University of Liverpool)

Title: **Invariant sets from Kerekjarto/Birkhoff**

Time: Thursday February 27, 2020, 13:00-14:00

Place: Room MATH-117

Abstract: Bela Kerekjarto was a Hungarian topologist active in the 1930's. He is well-known in some quarters, but I first came across him while researching the work of Yael Naim Dowker, who taught at Imperial from the mid 1950's until her retirement in the mid 1980's. Some of her work, in particular with her Ph D student George Lederer, impinged on topological dynamics. Their joint work used a remarkable result which she attributed to Kerekjarto, which in essence shows the existence of nontrivial stable/unstable sets for any compact invariant set, some 25 years before the publication of the now-classical corresponding results for the differentiable category. I shall discuss the implications for the structure of the set of invariant sets of a homeomorphism of a compact metric space. There will be more questions than answers.

Speaker: **Toby Hall** (University of Liverpool)

Title: **Natural extensions of unimodal maps are virtually sphere homeomorphisms**

Time: Thursday February 13, 2020, 13:00-14:00

Place: Room MATH-117

Abstract: Let f be a unimodal map of the interval. The inverse limit construction replaces f with a self-homeomorphism - the natural extension of f - of its inverse limit space. These inverse limit spaces have complicated and intricate topology: for example, they typically contain indecomposable continua.

I'll discuss recent work with Philip Boyland (University of Florida) and André de Carvalho (University of Sao Paulo), in which we construct mild semi-conjugacies from these natural extensions to sphere homeomorphisms. The family of sphere homeomorphisms constructed in this way includes examples of Thurston's pseudo-Anosov maps; of generalized pseudo-Anosovs; and of (further generalized) measurable pseudo-Anosovs.

Speaker: **Tania Gricel Benitez Lopez** (University of Liverpool)

Title: **Julia continua of transcendental entire functions**

Time: Thursday January 30, 2020, 13:00-14:00

Place: Room MATH-117

Abstract: Recently Rempe-Gillen gave an almost complete description of the possible topology of the Julia continua of disjoint-type functions combining well-studied concepts from continuum theory and new techniques of transcendental dynamics. In particular, he constructed a function where all Julia continua are pseudo-arcs, however this arises as a special case of a more general construction. In this presentation, we discuss how to construct a disjoint-type function such that all Julia continua are pseudo-arcs using a different technique which is more explicit, and as a result we obtain better control over the lower order of growth of the function.

Speaker: **Leticia Pardo Simón** (Polish Academy of Sciences)

Title: **Entire functions whose maximum modulus set has prescribed discontinuities**

Time: Thursday November 21, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: In 1909, Hardy gave an example of a transcendental entire function f so that the set of points where f achieves its maximum modulus, M(f), has infinitely many discontinuities. This is one of only two known examples of such a function. In this talk, we will significantly generalise these examples. In particular, we will show that, given an increasing sequence of positive real numbers, tending to infinity, there is a transcendental entire function, f, such that M(f) has discontinuities with moduli at all these values. This is joint work with Dave Sixsmith.

Speaker: **Gabriel Fuhrmann** (Imperial College)

Title: **Unique ergodicity and zero entropy of irregular almost automorphic symbolic extensions of irrational rotations and applications to quasicrystals**

Time: Thursday November 14, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: A classical result by Markley and Paul states that irregularalmost automorphic symbolic systems over irrational rotations are typically not uniquely ergodic and have positive entropy. By constructing a particular Cantor set, we prove that for each irrational rotation there still are almost automorphic extensions which are mean-equicontinuous (and hence have zero entropy and are uniquely ergodic).

The talk aims to be accessible by a broad audience. To that end, we will first motivate the study of almost automorphic systems by taking a look at well-studied representatives: dynamical systems associated to quasicrystals (i.e. Delone dynamical systems). After a short review of the results of Markley and Paul, we will eventually discuss the construction of the above mentioned Cantor set.

This is a joint work with Eli Glasner, Tobias Jäger and Christian Oertel.

Speaker: **James Waterman** (University of Liverpool)

Title: **Wiman-Valiron discs and the Hausdorff dimension of Julia sets of meromorphic functions**

Time: Thursday November 7, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: The Hausdorff dimension of the Julia set of transcendental entire and meromorphic functions has been widely studied. We review results concerning the Hausdorff dimension of these sets starting with those of Baker in 1975 and continuing to recent work of Bishop. In particular, Baranski, Karpinska, and Zdunik proved that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic function with a particular type of domain called a logarithmic tract is greater than one. We discuss generalizing this result to meromorphic maps with a simply connected direct tract and certain restrictions on the singular values of these maps. In order to accomplish this, we develop tools from Wiman-Valiron theory, showing that some tracts contain a dramatically larger disk about maximum modulus points than previously known.

Speaker: **Charles Walkden** (University of Manchester)

Title: **Stability index and Weierstrass functions**

Time: Thursday October 31, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: A dynamical system may have multiple attractors. The basin of attraction for a given attractor is the set of points which, under interation, converge to that attractor. In the case where there are multiple attractors, the basins may have very complicated local structure, for example the boundary between them could be highly irregular. The stability index (introduced numerically by Alexander-Yorke-You-Kan, Sommerer-Ott and others in physical systems in the early 1990s and studied more theoretically by Podvigina-Ashwin (2011) and Keller (2015, 2017)) is a number, behaving like a local dimension, that captures the complexity of the local structure of the basins of attraction. This talk will discuss how one can explicitly calculate the stability index in the case of skew-product dynamical systems, using classical tools and techniques from hyperbolic dynamics and thermodynamic formalism.

Speaker: **Thomas Kecker** (University of Portmouth)

Title: **Complex differential equations and their singularities**

Time: Thursday October 24, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: We will consider certain classes of 2nd-order non-linear, non-autonomous, ordinary differential equations in the complex domain and discuss what types of singularities the solutions of these can develop when analytically continued. In particular, we will study the local and global behaviour of the solutions in the complex plane. Some of the classes of equations discussed include the famous Painlevé equations, which have found many applications in mathematical physics recently.

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

Title: **Dynamics of transcendental entire functions with escaping singular orbits**

Time: Monday October 14, 2019, 14:00-15:00

Place: Room MATH-G16

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

Title: **Building surfaces from equilateral triangles**

Time: Thursday October 10, 2019, 13:00-14:00

Place: Room MATH-117

Abstract: In this talk, we consider the following natural question. Suppose that we glue a (finite or infinite) collection of equilateral triangles together in such a way that each edge is identified with precisely one other edge, each vertex is identified with only finitely many other vertices. If the resulting surface is connected, it naturally has the structure of a Riemann surface, i.e., a one-dimensional complex manifold. We ask which surfaces can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface can arise from this construction if and only if it is defined over a number field. These Belyi surfaces and their associated “Belyi functions” have found applications across many fields of mathematics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces).

Speaker: **Daniel Meyer** (University of Liverpool)

Title: **Uniformization of quasitrees**

Time: Thursday October 3, 2019, 13:00-14:00

Place: Room MATH117

Abstract: A quasisymmetry maps balls in a controlled manner. These maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider "quasitrees''. We show that any such tree is quasisymmetrically equivalent to a geodesic tree. Under additional assumptions it is quasisymmetric to the "continuum self-similar tree''. This is joint work with Mario Bonk.