# Topology and Dynamics seminar

## In 'Topology and Dynamics'

Back to 'Mathematics Research'Welcome to the Topology and Dynamics seminar homepage. If you would like to be added to the mailing list for this seminar series, then please email the seminar organisers: Andrew Mitchell or Tony Samuel.

## A naturally appearing family of Cantorvals

A Cantorval is a subset of the real line enjoying properties both of an interval and a Cantor set. In the talk, we show the existence of a large family of Cantorvals arising in the projection description of primitive two-letter substitutions.

- Speaker: Jan Mazac (Bielefeld University, Germany)
- Wednesday 03 May 2024, 13:00-14:00
- Watson Building, Lecture Theatre C

## Graphical models for infinite measures with applications to extremes and Lévy processes

Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure on the punctured Euclidean space that explodes at the origin. The importance of such measures stems from their connection to infinitely divisible and max-infinitely divisible distributions, where they appear as Lévy measures and exponent measures, respectively. We characterize independence and conditional independence for such a measure in various ways through kernels and factorization of a modified density, including a Hammersley-Clifford type theorem for undirected graphical models. As opposed to the classical conditional independence, our notion is intimately connected to the support of the measure. Our general theory unifies and extends recent approaches to graphical modeling in the fields of extreme value analysis and Lévy processes.

- Speaker: Kirstin Strokorb (Cardiff University, UK)
- Thursday 02 May 2024, 14:00-15:00
- Watson Building, Lecture Theatre B

## A cornucopia of bounds

In this talk we consider right-infinite words over a finite alphabet, which are generated via substitution rules. A well-known way of studying complexity of these words is via the question 'how many finite subwords of a given length does this infinite word have?', which gives rise to the notion of subword complexity. If instead one considers what types of (finite) subwords occur as arithmetic subsequences, one can obtain a different and very interesting measure of complexity. In this talk, we consider the occurrence of monochromatic (i.e. same letter) arithmetic progressions within right-infinite words, and provide asymptotic growth rates in some (general) cases. No previous knowledge is assumed, and the talk will start from the basics of substitution systems.

- Speaker: Petra Staynova (University of Derby, UK)
- Thursday 07 March 2024, 14:00-15:00
- Watson Building, Lecture Theatre B

## Self-similarity of substitution tiling semigroups

Substitution tilings arise from graph iterated function systems. Adding a contraction constant, the attractor recovers the prototiles. On the other hand, without the contraction one obtains an infinite tiling. In this talk I'll introduce substitution tilings and an associated semigroup defined by Kellendonk. I'll show that this semigroup defines a self-similar action on a topological Markov shift that's conjugate to the punctured tiling space. The limit space of the self-similar action turns out to be the Anderson-Putnam complex of the substitution tiling and the inverse limit recovers the translational hull.

This was joint work with Jamie Walton.

- Speaker: Mike Whittaker (University of Glasgow, UK)
- Thursday 29 February 2024, 14:00-15:00
- Watson Building, Lecture Theatre B

## Open problems on Minkowski measurability, Weyl laws, self-affine sets, Rauzy fractals and the Riemann hypothesis

We will discuss some open (and some solved) problems relating Minkowski measurability and Weyl laws to the Riemann hypothesis. In this realm, it is particularly interesting to study fractal sets on the real line that are more general than self-similar sets whose associated IFS satisfies the OSC. In higher dimensions some results are known for self-similar and self-conformal sets. In this talk, we will address these questions for Rauzy fractals and self-affine sets.

- Speaker: Sabrina Kombrink (University of Birmingham, UK)
- Thursday 22 February 2024, 14:00-15:00
- Watson Building, Lecture Theatre B

## The fractal, the curved, and the well-approximable

Given a self-similar set $F$, a manifold $M$ and a set of well-approximable points $W$ in $\mathbb{R}^k$, Marstrand slicing theorem gives us a 'generic' upper bound that $\dim (F\cap M\cap W)\leq \max\{\dim F+\dim M+\dim W-2,0\}$ which can be conjectured as an actual inequality under some non-resonance conditions among $F,M,W.$ We show that this is the case for $F$ being a large enough missing-digit Cantor set, $M$ being a hypersurface with non-vanishing Gaussian curvature and $W=W_\nu$ whose Diophantine exponent is bigger than but close to the trivial exponent $\nu=1/k$.

This is a joint work with S. Chow.

- Speaker: Han Yu (University of Warwick, UK)
- Thursday 15 February 2024, 14:00-15:00
- Watson Building, Lecture Theatre B

## Fourier decay for fractal measures and their pushforwards

Determining when the Fourier transform of a measure decays to zero as a function of the frequency, and estimating the rate of decay if so, is an important problem. We will discuss this problem in relation to fractal measures arising from iterated function systems, explaining that systems with non-linearity often result in good decay. In particular, we use a disintegration technique to prove that the Fourier transforms of non-linear pushforwards of a general class of fractal measures decay at a polynomial rate. Combining this with a result of Algom – Rodriguez Hertz – Wang and Baker – Sahlsten, we prove that for any IFS on the line consisting of analytic contractions, at least one of which is not affine, every non-atomic self-conformal measure exhibits polynomial Fourier decay. This has applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, and normal numbers in fractal sets.

This talk is based on recent joint work with S. Baker.

- Speaker: Amlan Banaji (Loughborough University, UK)
- Thursday 08 February 2024, 14:00-15:00
- Old Gym, LG06

## Scaling properties of (generalised) Thue-Morse measures

The Thue-Morse measure and its generalisations are diffraction measures of simple aperiodic systems. Besides that, they are paradigmatic examples of purely singular continuous probability measures on the unit interval given as an infinite Riesz product. To study their scaling behaviour a classical method, the thermodynamic formalism can be used – which however has to be adapted to an unbounded potential. Besides seeing this method, we will also see how quantitatively the Birkhoff and dimension spectrum changes depending on the point of the singularity. This is joint work with M. Baake, P. Gohlke, and M. Kesseböhmer.

- Speaker: Tanja Schindler (Jagiellonian University, Poland)
- Thursday 11 January 2024, 15:00-16:00
- Watson Building, Lecture Theatre C

## An Abstraction of the Unit Interval with Euclidean Topology and Denominators

Compact Hausdorff spaces are the topological abstraction of the unit interval [0,1] (in a sense that can be made precise). Let us now equip the unit interval with the “denominator map” den: [0,1] → N that maps a rational number to its denominator and an irrational number to 0. We characterize the abstraction of [0,1] that takes into account both the topology and the denominator map. The reason why we were interested in this problem is that one can show that the resulting structures form a category that is categorically dual to the category of Archimedean metrically complete Abelian lattice-ordered groups. This is a joint work with V. Marra and L. Spada.

- Speaker: Marco Abbadini (Computer Science, University of Birmingham)
- Thursday 23 November 2023, 15:00-16:00
- LG10 Old Gym

## The Troublesome Probabilistic Powerdomain

The probabilistic powerdomain was introduced by Jones and Plotkin in 1989 and some fundamental properties were proved in the PhD thesis of Jones. Several authors were able to establish a close link between this construction and Borel measures on fairly general topological spaces. From the point of view of semantics of programming languages, however, one would like to know whether the construction can be restricted to one of the Cartesian closed categories of continuous domains. This remains an open problem. In this talk I will present some of the background and some of the recent progress towards a positive solution.

- Speaker: Achim Jung (University of Birmingham)
- Thursday 09 November 2023, 15:00-16:00
- LG10 Old Gym

## Non-Hausdorff Topology and Mathematical Analysis

In classical mathematical analysis, the topologies that are commonly used are Hausdorff. In contrast, over partial orders, the topologies that capture the concept of approximation are typically non-Hausdorff. Continuous domains are a special class of partial orders that were introduced by Dana Scott (in the late sixties) as a mathematical model of computation. Domain theory has enriched computer science with powerful methods from order theory, topology, and category theory, and domains are ideal for analysis of robustness, soundness, completeness, and computability.

Many concepts of mathematical analysis have been recast in domain theory, e.g., dynamical systems, iterated function systems, measure theory, non-smooth analysis, stochastic processes, and solution of ODEs. The aim of the talk is to present a brief introduction to domain-theoretic mathematical analysis together with some key results. This will be followed by a discussion of some open problems, e.g., solution of PDEs. A common challenge in addressing these problems is handling topological spaces which do not have favourable properties (e.g., local compactness, core-compactness, etc.) that are necessary for domain constructions. An overview of recent results in this area will be presented as well.

- Speaker: Amin Farjudian (School of Mathematics, University of Birmingham)
- Thursday 26 October 2023, 15:00-16:00
- LG10 Old Gym

## Cut and Project sets and distances in self-similar sets

The main point of this talk will be to define a family of cut and project sets which are of interest in number theory and fractal geometry. I’ll link some old questions in Fractal geometry to the properties of these cut and project sets.

- Speaker: Tom Kempton (University of Manchester)
- Thursday 18 May 2023, 15:00-16:00
- Watson Building, Room 310

## Epilepsy: A dynamical systems perspective

To quote Richard Feynman, “Linear systems are important because we can solve them.” But the human brain is anything but linear and fathomable. So, how can we understand it?

As is the practice in nonlinear dynamics, we will confront the system in question, perturb it, play with it, and see how it responds. I will start with assumption that the brain is a complex, nonlinear system and play with its inputs, outputs and parameters and try to form a convincing story of how the brain could transition into a seizure and back. I will also pose some open questions I have encountered in this journey:

(1) What can we infer about the statistical properties of the ensemble of neurons making up the brain, when all that we could measure using electroencephalography (EEG) is an aggregated form of the ensemble response?

(2) What causes epileptic seizures to terminate?

(3) Can Melnikov methods be extended to study the switching between the limit cycle and fixed point attractors in a Hopf bifurcation?

I will present the context of these questions and expand on them further during the talk, with the hope of exploring answers together.

- Speaker: Aravind Kumar Kamaraj (University of Birmingham)
- Thursday 11 May 2023, 15:00-16:00
- Watson Building, Room 310

## Lipschitz images of the Cantor set

The Analyst’s Traveling Salesman Problem is to characterize those sets that can be covered by a Lipschitz image of [0,1]. The problem we get by replacing the interval by the Cantor set might be called the Fractal Analyst’s Traveling Salesman Problem. Another motivation comes from the well known classical result that the compact metric spaces are exactly the continuous images of the Cantor set, so it seems to be natural to ask which metric spaces can be obtained as a Lipschitz image of the Cantor set.

We prove that every compact metric space of upper box dimension less than log2/log3 can be obtained as the Lipschitz image of the Cantor set. We characterize those self-similar sets with the strong separation condition that can be obtained as the Lipschitz image of the Cantor set. In fact, we prove more general results than these ones and we also have other results that we needed or obtained as a spin off. Among others we show that in some sense every reasonable fractal dimension must be at least the Hausdorff dimension and at most the upper box dimension and we give a characterization of those compact metric spaces that can be obtained as an alpha-Hölder image of [0,1]. This is joint work with Richárd Balka.

- Speaker: Tamas Keleti (Eötvös Loránd University, Hungary)
- Thursday 20 April 2023, 13:00-14:00
- Teaching and Learning Building, Room 119

## Spectral properties of random hyperbolic surfaces

The aim of this talk is to present recent progress in the spectral theory of hyperbolic surfaces, which was achieved by bringing a new probabilistic viewpoint to this classic field. Studying random hyperbolic surfaces allows to use averaging techniques and hence obtain results which are out of reach by deterministic means. The challenge here is to find a reasonable probabilistic model, that allows for computations. In this talk, after introducing and motivating the Weil—Petersson probabilistic setting, I will present estimates on the density of eigenvalues of the Laplacian and the spectral gap of typical surfaces.

- Speaker: Laura Monk (University of Bristol)
- Thursday 13 April 2023, 15:00-16:00
- Arts, Room 315

## From non-integer expansions to fractal intersections

In this lecture we set up a bijection between the number of points in the horizontal intersection of a family of fractal curves and the number of orbits of a point in an augmentation of the base-q dynamics for some q in the interval (0,1). Key results from non-integer expansions are shown to have natural extensions giving some low-hanging fruit we can use to study fractal intersections. Time permitting, we will also expose some of the new results we have found on this topic.

- George Bender (University of Birmingham)
- Thursday 26 January 2023, 15:00-16:00
- Watson Building, Room 310

## An introduction to Rauzy fractals

Rauzy fractals are geometric objects that arise from substitutions that are the attractor of a graph-directed iterated function system that reflects the underlying substitution action. In this talk, I will provide an introduction to substitutions and Rauzy fractals and outline some of their key topological and dynamical properties. Time permitting, I will also discuss tiling properties of Rauzy fractals and how these are related to the Pisot conjecture.

- Speaker: Andrew Mitchell (University of Birmingham)
- Thursday 17 November 2022, 15:00-16:00
- Watson Building, Room 310

## On Poincaré inequalities and counting functions on 1-foliated domains with fractal boundaries (Part II)

The basic necessary notions to understand the variational formulation of eigenvalue problems of a Laplace operator with Neumann boundary conditions are introduced. We summarise some known results (in particular the Weyl-Berry conjecture and related results). A constructive method to obtain estimates for Poincaré constants on certain domains with fractal boundary is developed by introducing specific 1-foliations on the domain. This is used to estimate eigenvalue counting functions for domains with rough boundary and in particular for snowflake-like domains.

- Speaker: Lucas Schmidt (University of Birmingham)
- Thursday 03 November 2022, 15:00-16:00
- Watson Building, Room 310

## On Poincaré inequalities and counting functions on 1-foliated domains with fractal boundaries (Part I)

The basic necessary notions to understand the variational formulation of eigenvalue problems of a Laplace operator with Neumann boundary conditions are introduced. We summarise some known results (in particular the Weyl-Berry conjecture and related results). A constructive method to obtain estimates for Poincaré constants on certain domains with fractal boundary is developed by introducing specific 1-foliations on the domain. This is used to estimate eigenvalue counting functions for domains with rough boundary and in particular for snowflake-like domains.

- Speaker: Lucas Schmidt (University of Birmingham)
- Thursday 27 October 2022, 14:00-15:00
- Arts Building, Room 237

## Patterns in sets of fractional dimension and harmonic analysis in non-linear dynamics

It goes to the classical works of Szemerédi et al. to study the size of subsets of integers avoiding linear or non-linear progressions. Analogous problem of this in the real line goes back to the works of Bourgain and several subsequent papers, where for sets of zero Lebesgue measure one typically imposes some Fourier analytic assumptions on the set (e.g. the existence of a Rajchman measure whose Fourier decay rates relates closely to the Frostman properties of the measure). In fact, the works of Keleti and Shmerkin show that some assumptions are necessary always in this regime to find arithmetic progressions, but it is not clear what is the exact assumption needed from the set to find certain patterns. The examples here are characteristic to linear patterns, so it is plausible for non-linear patterns one might not need any assumptions. Recently, in a joint work with B. Kuca and T. Orponen, we considered a non-linear Sárközy-type problem in the plane, where we did not need any assumptions. Instead, we exploited the non-linearity of the pattern by connecting it to a kind of spectral gap property of “parabolic minimeasures”. This still leaves open the question of finding linear progressions. Is there some more geometric assumption needed to always find arithmetic progressions in a set of fractional dimension? In a recent joint work with C. Stevens we studied this problem in the specific case of attractors of *C*^{2} dynamical systems, and connected the Fourier analytic assumptions to suitable non-linearity of the dynamics. In this talk we will attempt to give a very gentle and general introduction to this field and the methods we use.

- Speaker: Tuomas Sahlsten (Aalto University)
- Wednesday 15 June 2022, 15:00-16:00
- Watson Building, Room 310

## Random substitutions, Rauzy fractals and the Pisot conjecture

A substitution is an iterative rule for replacing letters with words, like a↦ab, b↦a, the Fibonacci substitution. The Pisot property means that all of the eigenvalues of the associated incidence matrix lie in the open unit disc, except for the leading eigenvalue, which is real and greater than 1. The Pisot conjecture can be formulated in terms of a related fractal called the Rauzy fractal, whose generating IFS is ‘dual’ to the substitution. In this talk, we will introduce these concepts and explain how a new approach that introduces randomness may help to circumvent some of the biggest obstacles usually encountered when tackling the Pisot conjecture.

This is based on joint work with P. Gohlke, A. Mitchell and T. Samuel.

- Speaker: Dan Rust (The Open University)
- Monday 16 May 2022, 14:10-15:10
- Watson Building, Room 310

## Random Lochs Theorem

Lochs’ theorem from 1964 concerns the number of continued fraction digits of a real number one can determine from knowing the first *n* decimal digits of this number. In 1999 Bosma, Dajani and Kraaikamp generalised this result to other number systems and placed it in a dynamical framework, where the statement relates the sizes of the cylinder sets of the systems involved. In this talk we will extend Lochs’ theorem to random dynamical systems and provide various examples of random number systems to which it applies.

This is based on joint work with E. Verbitskiy and B. Zeegers.

- Speaker: Charlene Kalle (Leiden University)
- Monday 16 May 2022, 13:10-14:10
- Watson Building, Room 310

## Measures arising from regular sequences: existence, ergodicity, and spectral classification

In this talk, we will discuss how to build probability measures from regular sequences, which are generalisations of automatic sequences which are not restricted to be bounded. We provide sufficient conditions which guarantee the existence of the measures and their ergodicity under a suitable subgroup of the torus. We also provide a complete classification of their Lebesgue spectral type (pure point, absolutely continuous, singular continuous) dependent on quantities derivable from the linear representation of the sequence (joint spectral radius and Lyapunov exponents).

This is based on joint works with M. Coons, J. Evans, and Z. Groth.

- Speaker: Neil Manibo (Open University)
- Thursday 05 May 2022, 15:00-16:00
- Watson Building, R17/18

## Dimensions of random cookie-cutter-like sets

The random cookie-cutter-like sets are defined as the limit sets of a sequence of random cookie cutter mappings. By introducing the weak Gibbs-like measures, we study the fractal dimensions of these random sets and show that the Hausdorff dimension, the packing dimension and the box-counting dimension coincide and are almost surely equal to the unique zero of the topological pressure function.

- Speaker: Wafa Ben Saad (Universität Bremen)
- Thursday 10 March 2022, 15:00-16:00
- Onloine

## On some recent results for the Lorentz gas with infinite horizon

In the first part of the talk I will recall recent local limit results including the very recent result with Melbourne and Pene on local large deviation. In the second part, I recall the set up of Lorentz gases with scatterer size going to zero. In joint work with Balint and Bruin, we obtain a precise understanding of the allowed path of taking limits as the size of the scatterers goes to zero and time goes to infinity so that the Central Limit Law (with non-standard normalization) persists. This type of joint law is, at some extent, in the gist of Boltzmann–Grad limit. Further, we obtain joint local limit theorems and mixing as the size of the scatterers goes to zero and time goes to infinity and I will comment on these.

- Speaker: Dalia Terhesiu (Universiteit Leiden)
- Thursday 03 March 2022, 15:00-16:00
- Online

## Fourier transforms and nonlinear dynamics

Currently there is a great deal of research being produced in the area of Fourier transforms for measures. There is particular interest in when such objects exhibit (polynomial) decay. Much of this research in the context of nonlinear-map invariant measures stems from the ground-breaking paper of Bourgain–Dyatlov (2017) on Patterson–Sullivan measures for Hyperbolic Surfaces. In this talk we will discuss some of the methods used in this paper, and how large deviation theory can be used to apply their main tool of Discretized Sum-Product theory to the setting of expanding nonlinear maps rather than Fuchsian groups. We will conclude by looking at the assumptions necessary to use the Sum-Product theory, and heuristically how to prove these assumptions.

- Speaker: Connor Stevens (University of Manchester)
- Thursday 10 February 2022, 15:00-16:00
- Online

## Notions of disorder for random substitutions

Substitutions are a classic tool to produce self-similar structures with a long-range order. The corresponding shift-dynamical systems can be deemed to be relatively ordered as they have vanishing entropy, and are often isomorphic to a rotation on a locally compact abelian group. By locally randomizing the substitution rule, we obtain dynamical systems that are much more disordered in a measure-theoretic, topological and combinatorial sense. At the same time, under appropriate conditions, they maintain long-range correlations, presenting themselves in a non-trivial pure point part of the diffraction measure. In this talk, we will discuss how to assess several notions of disorder in the context of random substitutions, including quantitative bounds for topological and measure-theoretic entropy.

This talk is based on joint work with A. Mitchell, D. Rust, T. Samuel, and T. Spindeler.

- Speaker: Philipp Gohlke (Universität Bielefeld)
- Thursday 09 December 2021, 15:00-16:00
- Online

## Complexity results for β-expansions

β-expansions are a simple generalisation of the well known binary/ternary/decimal representations of real numbers. One of the most interesting properties they have is that a real number *x* almost surely has uncountably many β-expansions. This means one can ask interesting questions about the size and complexity of the set of expansions. In this talk I will survey some results on the complexity of this set. The goal of the talk is to give an introduction to this topic and to the techniques involved. I will also pose some open questions.

- Speaker: Simon Baker (University of Birmingham)
- Thursday 25 November 2021, 15:00-16:00
- Online

## Geometric functionals of fractal percolation

Fractal percolation is a family of random self-similar sets suggested by Mandelbrot in the seventies to model certain aspects of turbulence. It exhibits a dramatic topological phase transition, changing abruptly from a dust-like structure to the appearance of a system spanning cluster. The transition points are unknown and difficult to estimate, and beyond the fractal dimension not so much is known about the geometry of these sets. It is a natural question whether geometric functionals such as intrinsic volumes can provide further insights.

We study some geometric functionals of the fractal percolation process *F*, which arise as suitably rescaled limits of intrinsic volumes of finite approximations of *F*. We establish the almost sure existence of these limit functionals, clarify their structure and obtain explicit formulas for their expectations and variances as well as for their finite approximations. The approach is similar to fractal curvatures but in contrast the new functionals can be determined explicitly and approximated well from simulations. Joint work with M. Klatt.

- Speaker: Steffen Winter (KIT)
- Thursday 18 November 2021, 15:00-16:00
- Online

## Path-dependent shrinking target problem in generic affine iterated function systems

The shrinking target problem studies the Hausdorff dimension of the set of points in a metric space whose orbits under a transformation hit a family of shrinking balls infinitely often. We consider a variation of such a problem by allowing the radii of the shrinking balls to depend on the path of the point itself in affine iterated function systems. It turns out that generically the Hausdorff dimension of this path-dependent shrinking target set is given by the zero point of a certain limsup pressure function. This is a joint work with H. Koivusalo and M. Rams.

- Speaker: Lingmin Liao (LAMA, UPEC)
- Thursday 11 November 2021, 15:00-16:00
- Online

## Computational and dynamic complexity in shift spaces

We will discuss some results from the article *The relationship between word complexity and computational complexity in subshifts* by R. Pavlov and P. Vanier, as well as related papers. We will not assume any background on Turing reducibilty or degrees.

- Speaker: Robert Leek (University of Birmingham)
- Thursday 28 October 2021, 15:00-16:00
- Online

## Toral Anosov diffeomorphisms with computable resonances

In the one-dimensional setting, Blaschke products give rise to analytic expanding circle maps for which the entire spectrum of the (compact) transfer operator is computable. Inspired by these examples, in this talk we will present a class of Anosov diffeomorphisms on the torus, constructed using Blaschke factors, for which the spectrum of transfer operators defined on a suitable anisotropic Hilbert spaces can be determined explicitly and related to the dynamical features of the underlying maps.

- Speaker: Julia Slipantschuk (University of Warwick)
- Thursday 21 October 2021, 15:00-16:00
- Online

## Transfer operators and BV spaces: from classic to anisotropic

Smooth ergodic theory aims to analyse the long-term statistics of chaotic dynamical systems. There are several analytic and probabilistic tools that are used to answer such questions. Each of these approaches has its advantages and its shortcomings, depending on the system under consideration. In this presentation, I will focus on transfer operator techniques and spectral methods, which are known to be very powerful when dealing with uniformly expanding, or uniformly hyperbolic systems. The first half of this talk will be rather elementary, aimed at non-experts, focusing on ideas behind this approach through simple, yet important examples. In the second half of the talk, I will discuss a recent joint work with C. Liverani, whose long-term goal is to provide a good spectral picture for piecewise hyperbolic systems with singularities (e.g. billiard maps). I will also discuss a recent joint work with F. Sélley on coupled map lattices.

- Speaker: Wael Bahsoun (Loughborough University)
- Thursday 27 May 2021, 15:00-16:00
- Online

## Multidimensional continued fractions and symbolic codings of toral translations

The aim of this lecture is to find good symbolic codings for translations on the *d*-dimensional torus that enjoy the well-known and nice properties of Sturmian sequences (as for instance low complexity and good local discrepancy properties, i.e., bounded remainder sets of any scale). Inspired by the approach of G. Rauzy we construct such codings by the use of multidimensional continued fraction algorithms that are realized by sequences of substitutions.

- Speaker: Jörg Thuswaldner (Montanuniversität Leoben)
- Thursday 20 May 2021, 15:00-16:00
- Online

## Thickness and intersections in **R**^{d}

We will talk about the connection between thickness and intersections in **R**^{d}. Since the study of the presence of homothetic copies of a given finite set (pattern) is related to the study of intersections, we will see as a consequence a result that guarantees patterns in thick compact sets.

This talk is partly based on a joint work with K. Falconer.

- Speaker: Alexia Yavicoli (The University of British Columbia)
- Thursday 06 May 2021, 15:00-16:00
- Online

## Counting asymptotics for fractals and renewal theory

We discuss several counting and geometric problems for fractals sets, which we will transform into a form to which newly developed renewal theory can be applied. Problems that we consider include counting the number of complementary intervals of restricted continued fraction digit sets exceeding a given length, counting the number of radii of circles in an Apollonian packing that exceed a given value, and determining the behaviour of the volume of the ε-parallel set of a fractal as ε tends to zero.

- Speaker: Sabrina Kombrink (University of Birmingham)
- Thursday 29 April 2021, 15:00-16:00
- Online

## Equidistribution results for self-similar measures

A well-known theorem due to Koksma states that for Lebesgue almost every *x* > 1 the sequence (*x ^{n}*) is uniformly distributed modulo 1. In this talk I will discuss an analogue of this statement that holds for fractal measures. As a corollary of this result we show that if

*C*is equal to the middle third Cantor set and

*t*≥ 1, then almost every

*x*in

*C*+

*t*is such that (

*x*) is uniformly distributed modulo 1. Here almost every is with respect to the natural measure on

^{n}*C*+

*t*.

- Speaker: Simon Baker (University of Birmingham)
- Thursday 22 April 2021, 15:00-16:00
- Online

## On Arithmetic Progressions Within Binary Words

We will present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Moreover, we will show how the arguments are inspired by van der Waerden’s proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

This talk is based on joint work with I. Aedo, U. Grimm, and Y. Nagai.

- Speaker: Petra Staynova (University of Derby)
- Thursday 15 April 2021, 15:00-16:00
- Online

## Random interval maps: Stationary measures and random matching

For a large class R of piecewise affine random systems of the interval we show how to obtain explicit formulas for the invariant densities of stationary measures. Next, we extend the notion of matching for deterministic transformations to random matching for random interval maps. We then prove that for systems in R the property of random matching implies that any invariant density is piecewise constant. We finally apply these results to a family of random maps producing signed binary expansions in order to study minimal weight expansions.

This talk is partly based on a joint work with K. Dajani and C. Kalle.

- Speaker: Marta Maggioni (Universiteit Leiden)
- Thursday 25 March 2021, 15:00-16:00
- Online

## When is the beginning the end?

Let *f* : *X* → *X* be a continuous map on a compact metric space *X* and let α_{f}, ω_{f}, and ICT _{f} denote the set of α‑limit sets, ω‑limit sets, and nonempty closed internally chain transitive sets respectively. α‑ and ω‑limit sets may be viewed as the beginnings and ends of orbit sequences. We show that if the map *f* has shadowing then every element of ICT _{f} can be approximated (to any prescribed accuracy) by both the α‑limit set and the ω‑limit set of a full-trajectory. In particular this means that the presence of shadowing guarantees that ᾱ_{f} = ῶ_{f} = ICT _{f} (where the closures are taken with respect to the Hausdorff topology on the space of compact sets). We progress by introducing a property which characterises when all beginnings are ends of all beginnings, and all ends, beginnings of all ends.

This talk is partly based on a joint work with C. Good and J. Meddaugh.

- Speaker: Joel Mitchell (University of Birmingham)
- Thursday 18 March 2021, 15:00-16:00
- Online

## Recognizability for sequences of morphisms

Given a measure preserving transformation T acting on a space X, *recognizability* is a combinatorial notion that ensures the existence of a sequence of generating partitions of the space X into towers, where the dynamics consists of moving up the tower. Recognizability is a notion that has proven very useful for substitution dynamical systems. In this talk we investigate extending this notion to S-adic dynamical systems, which are generalizations of substitution dynamical systems.

Let A be a finite alphabet and let A* be the monoid of all finite words on A. Whereas a substitution dynamical system is generated by iterating a single monoid morphism σ sending letters in A to words in A*, an S-adic system is generated by a sequence of morphisms.

We investigate different notions of recognizability for a monoid morphism σ: A → B*. Full recognizability occurs when each (aperiodic) point in B^{Z} admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ: A → A*, where the tiling must be compatible with the language of the substitution. We discuss conditions that ensure full recognizability.

Next we define recognizability and also eventual recognizability for sequences of morphisms which define an S-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable.

Finally we discuss how to apply recognizability to finding the eigenvalues of S-adic dynamical systems.

This is joint work with V. Berthé, P. Cecchi, W. Steiner and J. Thuswaldner.

- Speaker: Reem Yassawi (The Open University)
- Thursday 25 February 2021, 15:00-16:00
- Online

## Laplacians on fractals and Weyl’s asymptotics

This talk will present the speaker’s recent results on a geometrically canonical Laplacian on the limit sets of certain Kleinian groups which are round Sierpiński carpets (RSCs), i.e., subsets of **Ĉ** = **C** ∪ {∞} homeomorphic to the standard Sierpiński carpet with complement in **Ĉ** consisting of disjoint open disks in **Ĉ**. On the Apollonian gasket, Teplyaev (2004) had constructed a canonical Dirichlet form as one with respect to which the coordinate functions are harmonic, and the speaker later proved its uniqueness and an explicit expression of it in terms of the circle packing structure of the gasket. This last expression of the Dirichlet form makes sense on general circle packing fractals, including RSCs, and defines a geometrically canonical Laplacian on such fractals. Moreover, with the knowledge of some combinatorial structure of the fractal it is also possible to prove Weyl’s eigenvalue asymptotics for this Laplacian, which is of the same form as the circle-counting asymptotic formula by Oh and Shah [Invent. Math. 187 (2012), 1–35]. The proof of Weyl’s eigenvalue asymptotics is based on a serious application of Kesten’s renewal theorem [Ann. Probab. 2 (1974), 355–386] to a certain Markov chain, which can be considered as a skew product random dynamical system, in the space of all possible Euclidean shapes of the pieces of the fractal. A short exposition of these results can be found in arXiv:2001.07010, and a sketch of the application of Kesten’s renewal theorem can be found in arXiv:2001.11354.

If time permits, a possible approach toward extensions to the case of self-conformal fractals in **C** consisting of nowhere rectifiable curves will also be mentioned.

- Speaker: Naotaka Kajino (Kobe University, Japan)
- Thursday 18 February 2021, 15:00-16:00
- Online

## Dynamical behaviour of alternate base expansions

We consider a generalisation of the β-expansion by applying cyclically the bases β_{0},..., β_{p-1}. We refer to the resulting expansion as an (alternate base) β-expansion. Just as in the case of the classical β-expansion one has typically uncountably many expansions. We concentrate first on the greedy expansion, we introduce a dynamical system generating them and study its ergodic properties. We also compare the alternate base β-expansion with the greedy β_{p-1}... β_{0}-expansion with digits in some special set, and characterise when these expansions are the same. We end by introducing the lazy expansion and show that the dynamical system underlying such expansions is isomorphic to the greedy counterpart.

This is joint work with E. Charlier, and C. Cisternino.

- Speaker: Karma Dajani (Utrecht University)
- Thursday 11 February 2021, 15:00-16:00
- Online

## Laplacian eigenfunctions on large genus random surfaces

Within the mathematical physics literature, it is thought that the Laplacian operator should exhibit properties that depend solely upon the ambient geometry of the space. Compact hyperbolic surfaces offer a geometrically rich setting upon which one can develop these ideas, and they are interesting due to their connection with conjectures arising in quantum mechanics. In this talk, I will discuss some recent work with Laura Monk (Strasbourg) regarding certain geometric features of these surfaces that occur with high probability in an appropriate model for random surfaces. Using this, I will then demonstrate how such geometry implies certain non-localisation properties of eigenfunctions on typical large genus compact hyperbolic surfaces

This is joint work with C. Gilmore, E. Le Masson, and T. Sahlsten.

- Speaker: Joe Thomas (University of Manchester)
- Thursday 04 February 2021, 15:00-16:00
- Online

## Measure theoretic entropy of random substitution subshifts

Random substitutions and their associated subshifts provide a model for structures that exhibit both long range order and positive entropy. In this talk we discuss the entropy of a large class of ergodic measures, known as frequency measures, that arise naturally from random substitutions. We introduce a new measure of complexity, namely measure theoretic inflation word entropy, and discuss its relationship to measure theoretic entropy. We also show how this new measure of complexity can be used to provide a framework for the systematic study of the measure theoretic entropy of random substitution subshifts.

As an application of our results, we obtain closed form formulas for the entropy of a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we show that this measure is the unique measure of maximal entropy.

This is joint work with P. Gohlke, R. Leek, D. Rust, and T. Samuel.

- Speaker: Andrew Mitchell (University of Birmingham)
- Thursday 28 January 2021, 15:00-16:00
- Online

## An introduction to the transfer operator method

In this series of seminars I will present an introduction to the transfer operator method. The transfer operator encodes information about an iterated map and is used to study the behaviour of dynamical systems, with applications to, for example, the calculation of Lyapunov exponents and decay of correlation. In this first talk I will present the definition and key properties of the transfer operator, and calculate the transfer operator for some well-known dynamical systems.

- Speaker: Andrew Mitchell (University of Birmingham)
- Friday 06 March 2020, 15:00-16:00
- Watson Building, Lecture Theater C

## Equicontinuity, transitivity and sensitivity

Robert Devaney defined chaos as a sensitive, transitive map where the set of periodic points is dense in the phase space. With an elegant proof, Banks et al showed that the the latter two properties entail the first. Since then, various analogues and generalisations of this result have been offered. Central to these theorems lie the notions of transitivity, equicontinuity, minimality and sensitivity.

In this talk I take a topological approach to dynamics and discuss sensitivity, topological equicontinuity and even continuity. I will provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. Time permitting, I will define what it means for a system to be eventually sensitive and give a dichotomy for transitive dynamical systems in relation to eventual sensitivity.

This talk is based upon joint work with C. Good and R. Leek.

- Speaker: Joel Mitchell (University of Birmingham)
- Thursday 06 February 2020, 14:00-15:00
- Watson Building, Lecture Theater C

## Growth along geodesic rays in hyperbolic groups

Let G be a non-elementary hyperbolic group equipped with a finite generating set S. Suppose that G acts cocompactly by isometries on a space X. If we fix an origin for X then we can ask the following general question: by how much does a group element g in G displace the origin and, how does this displacement compare to the word length of g (with respect to S)? In this talk we will discuss one way of answering this question. More specifically we will study how the displacement of the origin grows as we travel along infinite geodesic rays in the Cayley graph of G.

- Speaker: Stephen Cantrell (University of Warwick)
- Friday 28 February 2020, 15:00-16:00
- Watson Building, Lecture Theater A

## New approaches to overlapping iterated function systems

A standard technique for generating fractal sets is to use an object called an iterated function system (or IFS for short). When an IFS satisfies some separation condition then much is known about the corresponding fractal. The situation is far more complicated when the IFS fails to satisfy any separation condition, and the pieces of the fractal overlap significantly. In this talk I will discuss a new approach for describing how an IFS overlaps. This approach uses ideas from Diophantine approximation.

- Speaker: Simon Baker (Birmingham)
- Wednesday 29 January 2020, 13:00-14:00
- Watson Building, Lecture Theater A

## Random substitutions and topological mixing

I’ll introduce a new class of symbolic dynamical system associated with ‘random substitutions’. They’re the positive-entropy cousins of substitution subshifts and so many of the techniques that are used to study substitutions can analogously be utilised in the random setting. I’ll explain how we are able to determine when one of these subshifts is topologically mixing by studying an appropriate abelianisation of the substitution and which cases are still unresolved.

Joint work with E. Provido, L. Sadun and G. Tadeo

- Speaker: Dan Rust (Bielefeld)
- Thursday 23 January 2020, 10:00-11:00
- Watson Building, Room R17/18

## Dimensions of exceptional self-affine sets in **R**R^{3}

Planar self-affine sets generated by diagonal and anti-diagonal matrices are an important family of exceptional self-affine sets, where the box (and Hausdorff) dimension can be strictly smaller than the affinity dimension. The box dimensions are given by a natural ‘pressure type’ formula based on modified singular value functions. We consider the analogous setting in R^3, where the self-affine sets are generated by generalised permutation matrices, and will see that the situation is rather more complicated.

This is joint work with N. Jurga (Surrey).

- Speaker: Jonathan Fraser, University of St Andrews
- Tuesday 12 November 2019, 11:00-12:00
- Watson Building, Room G23

## Specification and Synchronisation for unique expansions

Given a positive integer M and q in (1,M+1] we study expansions in base q for real numbers over the alphabet {0,...,M}. In particular, we study some dynamical properties of a natural occurring subshift related to unique expansions in such base q known as symmetric lexicographic subshift. During the talk we will give a characterisation of the set of q’s such that such subshift has the specification property as well as the set of q’s such that such subshift is synchronised. We relate our results to those shown by Schmeling in (Ergodic Theory and Dynamical Systems, 17:675—694, 6 1997) in the context of greedy expansions in base q.

- Speaker: Rafael Alcaraz Barrera (Universidad Autónoma de San Luis Potosí, Mexico)
- Thursday 05 September 2019, 11:30-12:30
- Watson Building, Lecture Theater C

## Value quantales and Lipschitz constants

In 1997 Flagg introduced the concept of a value quantale, a certain ordered structure distilling key properties of the structure of [0,\infty]. Allowing a metric space to take values in a value quantale rather than insisting on the distances landing in [0,\infty] bears a remarkable consequence: All topological spaces are metrisable. As a result, Flagg’s formalism offers a unification of topology and metric geometry. I will sketch the proof of the result and present recent results emanating from the unified perspective. In particular, a Lipschitz ‘machine’ will be presented; a functor from diagrams of value quantales that produces a category of spaces and mappings with a suitable notion of Lipschitz constant. The classical notion is recovered by running the machine on a certain simple diagram.

- Speaker: Ittay Weiss (University of Portsmouth)
- Thursday 16 May 2019, 13:00-14:00
- Watson Building, Lecture Theater C

## Tanaka continued fractions and matching

In this talk we will first show that, for the Ito Tanaka continued fractions, matching holds almost everywhere. We do so by exploiting a relation between the orbit of the endpoints of the domain. In the second part we will try to adapt this proof for another family of continued fractions named the (N,\alpha)-continued fractions. This turns out to be difficult. Can we overcome these difficulties? The first part is joint work with Wolfgang Steiner and Carlo Carminati. The second part is joint work with Cor Kraaikamp.

- Speaker: Niels Langeveld (Universiteit Leiden)
- Thursday 09 May 2019, 13:00-14:00
- Watson Building, Lecture Theater C

## Dynamics of isometries of the hyperbolic plane

The well-known Denjoy–Wolff theorem describes the behaviour of iterates of holomorphic self-maps of the unit disc. Our objective is to develop results of a similar type for nonautonomous dynamical systems, in which sequences are generated by composing holomorphic maps from some given family. We make good progress for families of hyperbolic isometries, relating the dynamical systems to semigroups of transformations generated by these families.

- Speaker: Ian Short (Open University, UK)
- Thursday 02 May 2019, 13:00-14:00
- Watson Building, Lecture Theater C

## Locally finite trees and the topological minor relation

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO (better-quasi-order). Naturally, two interesting questions arise:

1. What is the number \lambda of topological types of locally finite trees?

2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this talk we address both questions by showing – entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

- Speaker: Jorge Bruno (University of Winchester)
- Thursday 25 April 2019, 13:00-14:00
- Watson Building, Lecture Theatre C

## Fractal Weyl bounds

Resonances of Riemannian manifolds play an important role in many areas of mathematics, e.g. analysis, dynamical systems, mathematical physics, and number theory. We will discuss some recent results (obtained together with F. Naud and L. Soares) on the localization and distribution of resonances for certain hyperbolic surfaces of infinite area. No prior knowledge is assumed; in particular, we will review the definition of resonances and the motivation of their investigation.

- Speaker: Anke Pohl (Universität Bremen, Germany)
- Thursday 18 April 2019, 13:00-14:00
- Watson Building, Lecture Theater C

## Uniqueness of trigonometric series outside fractals

A subset F of [0,1] is called a set of uniqueness if trigonometric series are unique outside of F. Otherwise F is called a set of multiplicity. The uniqueness problem in harmonic analysis dating back to the fundamental works of Riemann, Cantor et al. concerns about classifications of sets of uniqueness and multiplicity. Typically one expects the sets of uniqueness to have ’non-chaotic’ / orderly features and where as the sets of multiplicity should be `chaotic’. This is highlighted by the theorem of Piatetski-Shapiro-Salem-Zygmund (1954) stating that middle lambda-Cantor sets is a set of uniqueness if and only if 1/lambda is a Pisot number (roughly speaking numbers whose powers approximate integers at an exponential rate).

In our work we attempt to characterise the multiplicity or uniqueness of a fractal F using the statistical/dynamical properties of the fractal F. In particular, we prove any non-lattice self-similar set is a set of multiplicity. We also establish an analogous result in higher dimensions for self-affine sets, where the non-lattice condition is replaced by the irreducibility of the subgroup defined by the linear parts of the contractions. The statistical theory we use is the renewal theory for random walks on Lie groups.

Joint work with J. Li (Bordeaux & Zürich).

- Speaker: Tuomas Sahlsten (University of Manchester)
- Monday 01 April 2019, 15:00-16:00
- Watson Building, Lecture Theatre B

## Simultaneous shrinking target problems for ×2 and ×3

We consider the simultaneous shrinking target problems for ×2 and ×3. We obtain the Hausdorff dimensions of the intersection of two well approximable sets and also of the set of points whose orbits approach a given point simultaneously for these two dynamical systems. This is a joint work with Lingmin Liao.

- Speaker: Bing Li (South China University of Technology, Guangzhou)
- Monday 01 April 2019, 14:00-15:00
- Watson Building, Lecture Theatre B

## Quenched decay of correlations for slowly mixing systems

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of 'quenched' correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in $[\alpha_0,\alpha_1]\subset (0,1)$ chosen independently with respect to a distribution $\nu$ on $[\alpha_0,\alpha_1]$ and show that the quenched decay of correlation is governed by the fastest mixing map in the family.

In particular, we prove that for every $\delta >0$, for almost every $\omega \in [\alpha_0,\alpha_1]^\mathbb{Z}$, the upper bound $n^{1-\frac{1}{\alpha_0}+\delta}$ holds on the rate of decay of correlation for Hölder observables on the fibre over $\omega$.

This is a joint work with W. Bahsoun and C. Bose.

- Speaker: Marks Ruziboev (Loughborough University)
- Thursday 14 March 2019, 13:00-14:00
- Watson Building, Lecture Theatre C

## On the Hausdorff Dimension of Bernoulli Convolutions

Bernoulli convolutions are a simple family of self-similar measures with overlaps. The problem of determining which parameters give rise to Bernoulli convolutions of dimension one has been studied since the 1930s, and is still far from being completely solved. For algebraic parameters, we show how to give an expression for the dimension of the Bernoulli convolution in terms of random products of matrices, allowing us to conclude that the Bernoulli convolution has dimension one in many examples where the dimension was previously unknown. The problem has close connections to some problems in random dynamical systems, and yet we don’t really know best how to exploit these connections. This is joint work with Shigeki Akiyama, De-Jun Feng and Tomas Persson.

- Speaker: Tom Kempton (University of Manchester)
- Thursday 28 February 2019, 13:00-14:00
- Watson Building, Lecture Theatre C

## The action of Fuchsian groups on complex projective space

A Fuchsian group is a discrete subgroup of hyperbolic isometries. In this talk we will restrict our attention to the case of discrete subgroups of SO_{0}(2,1) that act on the hyperbolic plane with finite area quotient. This means the action on the hyperbolic plane is properly discontinuous and the limit set of the action is the whole ideal boundary. We may embed SO_{0}(2,1) into SL(3,ℂ) in the obvious way and study its action on CP^{2}. In this talk I will explain the correct notion of limit set for such an action and I will describe the topology of this limit set and of its complement, the region of discontinuity. This is joint work with Angel Cano and Pepe Seade.

- Speaker: John Parker (University of Durham)
- Thursday 07 February 2019, 13:00-14:00
- Watson Building, Lecture Theatre C

## Classes of continua determined by classes of mappings

If M is a class of continuous maps, we say a continuum X is M-closed if every continuous map from a continuum onto X is in the class M. We give a brief history of the subject, starting with Andrew Lelek’s Houston seminar in the ‘70s, and finish with a study of the class of M-closed continua, where M consists of the co-existential maps.

- Speaker: Paul Bankston (Marquette University)
- Thursday 17 January 2019, 14:00-15:00
- Watson Building, Lecture Theatre C

## Classification of attractors

After reviewing some general results highlighting the interplay of topology and dynamics, we will examine some new results on the topological classification of attractors. These results depend crucially on the understanding of the underlying dynamics and generalise the classification of the classical solenoids.

- Speaker: Alex Clark (Queen Mary, University of London)
- Thursday 10 January 2019, 13:00-14:00
- Watson Building, Lecture Theatre C

We address the challenges facing society and the economy, from shedding light on the refugee crisis, to character education in schools, through to developing leaders in the NHS.