Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
Generalized semigroup dynamics and their C*-algebras
Diego Martinez
University of Munster
Dynamical systems, that is, actions of (discrete) groups on locally compact spaces, have (classically) been used to construct broad classes of C*-algebras that are both interesting and manageable. In this talk we will discuss how to generalize this construction to inverse semigroup (twisted) actions, which yield a much larger class of C*-algebras that, nevertheless, retains much of the structure provided by the generalized dynamical system. Time permitting, we will also discuss how certain properties of the action are remembered by the construction, particularly with regards to amenability.
Badly approximable vectors and putative counterexamples to the Littlewood conjecture
Faustin Adiceam
Université Paris-Est Créteil
Answering a question raised by Bugeaud, the talk will be concerned with the determination of the Hausdorff dimension of a natural set interpolating between the set of badly approximable vectors (which is known to have full dimension) and the set of putative counterexamples to the Littlewood conjecture (which is known to have zero Hausdorff dimension). This is obtained by the construction of a so- called generalised Cantor set based on the Ostrowski numeration system induced by a badly approximable number. Time permitting, a quantitative refinement of this result will be shown to provide an alternative to the multiplicative theory of badly approximable vectors (of which the Littlewood conjecture is a particular case). This is work in progress with S. Seuret (UPEC).
Hyperbolic lattice point counting in unbounded rank
Chris Lutsko
University of Zurich
Counting lattice points in balls is a classical problem which goes back to Gauss in the Euclidean setting. In the hyperbolic setting this corresponds to counting matrices of norm T in SL(n,Z). For n=2 the record belongs to Selberg in the early 1980s. In a recent paper with Valentin Blomer we extend Selberg's method to higher rank (n > 2) and thus improve on the best known bounds for the hyperbolic lattice point counting problem in higher rank. In the first half of this talk I will introduce the problem, summarize the history, and give a sketch of Selberg's method. Then in whatever time remains, I will give a sketch of the proof of Blomer and myself.
Finding product sets in some classes of amenable groups
Andreas Mountakis
University of Warwick
In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erdős about sumsets in large subsets of the natural numbers. In this talk, we will discuss an extension of the result of the previous authors to several important classes of amenable groups, including finitely generated virtually nilpotent groups, and all abelian groups (G,+) with the property that the subgroup 2G has finite index. This is joint work with Dimitrios Charamaras.
On the complexity and geometry of Sturmians and Random Substitutions
Tony Samuel
University of Birmingham
Aperiodic sequences and sequence spaces form prototypical mathematical models of quasicrystals. The most quintessential examples include subshifts of Sturmian words and substitutions, which are ubiquitous objects in ergodic theory and aperiodic order. Two of the most striking features these shift spaces have are that they have zero topological entropy and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions, and in stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy and exhibit uncountably many ergodic measures. Moreover, they have been shown to provide mathematical models for physical quasicrystals with defects.
We will begin by talking about subshifts generated by Sturmian words and ways to measure their complexity beyond topological entropy, and show how this measure of complexity can be used to build a classification via Jarník sets. We will build a bridge between these aperiodic sequence spaces to random substitutions, and present some recent dynamical results concerning their measure theoretical entropy and ways to visualise them. Namely, we will present a method to build a new class of Rauzy fractals.
A cornucopia of bounds
Petra Staynova
University of Derby
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.
Mahler equations for Zeckendorf numeration
Reem Yassawi
Queen Mary University
Fixed points of Pisot substitutions can be visualised as projections, via a precompact acceptance window, of cut and project schemes. Fixed points of constant length substitutions are projections, along a diagonal, of a two-dimensional array which is the sequence of coefficients of the expansion of a rational function of two variables. In an attempt to understand the relationship between these two results, we define generalised versions of equations of q-Mahler type, which fixed points of constant length-q substitutions satisfy. For simplicity I will focus on substitutions whose characteristic polynomial has the golden mean as leading root. We show that fixed points of these substitutions satisfy a Zeckendorf-Mahler equation, and conversely, that isolating Zeckendorf-Mahler equations generate such fixed points. This is joint work with Olivier Carton.
Distribution of resonances for Anosov maps on the torus
Julia Slipantschuk
University of Warwick
Eigenvalues of transfer operators, known as Pollicott-Ruelle resonances provide insight into the long-term behaviour of the underlying dynamical system, in particular determining its exponential mixing rates. In this talk, I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.
Multifractal analysis for self-affine measures and phase transitions.
Thomas Jordan
University of Bristol
One form of multifractal analysis looks at local scaling properties for measures. We will start by reviewing what is known for self-similar measures with suitable separation conditions. We will then look at simple self-affine models (based on diagonal matrices) and look at how things differ form the self-similar case and also what can still be preserved. In particular will be looking at examples where the spectrum can be discontinuous and be non-concave. This is joint work with Istvan Kolossvary and Alex Rutar.
Random Pisot substitutions and their Rauzy fractals
Dan Rust
The Open University
The Pisot conjecture is a long-standing open question concerning the dynamical spectrum of subshifts associated with Pisot substitutions. One formulation of the conjecture can be stated in terms of the so-called Rauzy fractal for the substitution, which is the attractor of a graph-directed iterated function system that can be written purely in terms of data coming from the substitution. Random substitutions generalise substitutions and can be used to interpolate between them, allowing properties of one substitution to inform us of properties of another. In this talk, I'll introduce Rauzy fractals for Pisot random substitutions and results concerning natural measures that can be defined on the attractor. By studying these measures, we're able to prove an analogue of a multi-tiling result for Rauzy fractals but in the random setting, hinting at potential new tools for tackling the Pisot conjecture using probabilistic methods.
Strange attractors and the Borel conjecture
John Hunton
Durham University
The classical Borel conjecture aims, under appropriate conditions, to show that the homeomorphism class of a manifold is determined by its homotopy type. This talk discusses to whether a comparable conjecture may be formulated, or even proved, for expanding attractors. It represents joint work with Alex Clark.
Random walks on groups, amenability and ratio limit theorems
Richard Sharp
University of Warwick
A famous result of Kesten from 1959 relates symmetric random walks on countable groups to amenability. Precisely, provided the support of the walk generates the group, the probability of return to the identity in 2n steps decays exponentially fast if and only if the group is not amenable. This led to many analogous “amenability dichotomies”, for example for the spectrum of the Laplacian of manifolds and critical exponents of discrete groups of isometries. I will present a version of the dichotomy for non-symmetric walks. I will also discuss a new ratio limit theorem for amenable groups. This is joint work with Rhiannon Dougall.
Aperiodic Order and linear repetitivity of cut and project sets
James Walton
University of Nottingham
In this talk I will introduce the theory of Aperiodic Order by showcasing some standard examples of aperiodic tilings, giving a brief overview of the field’s connection with other areas and explaining the two main construction methods: substitution and the cut and project method. In the latter, one cuts points of a lattice which fall into a thickening, by a ‘window’, of an irrational hyperplane (the ‘physical space’), and then projecting to the physical space to obtain an ordered but non-periodic point set. Many important examples may be constructed in this way, such as the vertices of the Penrose tilings and the Ammann–Beenker tilings, and the method has interesting connections to Number Theory. I will explain a recent result for Euclidean cut and project sets with polytopal windows that classifies those which are ‘the most ordered’, in the precise sense of being linearly repetitive (LR). The result is that LR is equivalent to an algebraic condition together with a Diophantine Approximation condition, of the physical space staying far from the lattice.
Which domains are wandering domains?
David Martí-Pete
University of Liverpool
For a transcendental entire (or meromorphic) function, a wandering domain is a connected component of the Fatou set which is not eventually periodic. Wandering domains are very diverse in terms of both their topology and their dynamics. Boc Thaler proved the surprising result that every bounded regular domain such that its closure has a connected complement is the wandering domain of some transcendental entire function. Inspired by this result, together with Rempe and Waterman, we were able to construct further examples of wandering domains, including those that form Lakes of Wada. In this talk, we will explore the question of what domains are wandering domains of some entire or meromorphic function. This talk is based on joint works with Rempe and Waterman, and with Rippon, Sixsmith and Stallard.
Classification and statistics of cut-and-project sets
Yotam Smilansky
University of Manchester
Cut-and-project point sets are constructed by identifying a strip of a fixed n-dimensional lattice (the "cut"), and projecting the lattice points in that strip to a d-dimensional subspace (the "project”). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well-known examples such as the Fibonacci chain and vertex sets of Penrose tilings. Dynamical results concerning the translation action on the hull of a cut-and-project set are known to shed light on certain properties of the point set itself, but what happens when instead of restricting to translations we consider all volume preserving linear actions?
A homogenous space of cut-and-project sets is defined by fixing a cut-and-project construction and varying the n-dimensional lattice according to an SL(d,R) action. In the talk, which is based on joint work with René Rühr and Barak Weiss, I will discuss this construction and introduce the class of Ratner-Marklof-Strömbergsson measures, which are probability measures supported on cut-and-project spaces that are invariant and ergodic for the group action. A classification of these measures is described in terms of data of algebraic groups, and is used to prove analogues of results about a Siegel summation formula and identities and bounds involving higher moments. These in turn imply results about asymptotics, with error estimates, of point-counting and patch-counting statistics for typical cut-and-project sets.
A packing exponent formula for the upper box-counting dimension of the Rauzy gasket
Benedict Sewell
Alfréd Rényi Institute
The Rauzy gasket is a self-projective fractal that is an important subset of parameter space in numerous contexts in dynamics, topology and combinatorics on words. Estimating its fractal dimensions, even showing that they are nontrivial, has proven to be no easy task, despite lots of recent attention on self-projective sets.
In this talk, we outline an elementary proof of a general "packing exponent" formula for the upper box-counting dimension of a self-projective attractor, and use this to show that the two most popular notions of dimension (Hausdorff and box-counting) coincide for the Rauzy gasket, also giving the best known lower bound for this quantity.
The Dynamics of the Fibonacci Partition Function
Tom Kempton
University of Manchester
The Fibonacci partition function R(n) counts the number of ways of representing a natural number n as the sum of distinct Fibonacci numbers. For example, R(6)=2 since 6=5+1 and 6=3+2+1. An explicit formula for R(n) was recently given by Chow and Slattery. In this talk we express R(n) in terms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of R(n). This talk should be accessible to all, no knowledge of dynamics or number theory will be assumed.
Characteristic Factors in Topological Dynamics
Donald Robertson
University of Manchester
In applications of ergodic theory to combinatorics one often needs to prove positivity of certain correlations involving measure preserving transformations. It has been known since work of Furstenberg that it suffices to verify positivity for a special class of measure-preserving transformation possessing a lot of additional structure. Recent work of Glasner, Huang, Shao, Ye and Weiss has developed an analogous machinery in the setting of topological dynamics. In this talk I will survey their main results. No background knowledge will be assumed.