University of Birmingham
Grimm Network Meeting 
21 March 2025

Lunch and Welcome

Bratby Bar, Staff House, University of Birmingham

Pierre Arnoux (Aix-Marseille Université, CNRS, Marseille, France)

Title: Extensions of substitutions

Abstract: Substitutions are best defined as morphisms of the free monoid, or as positive automorphisms of the free group with a fixed basis. In that setting, primitive substitutions define a finite number of infinite fixed words, and an associated dynamical system. But these fixed words can also be seen as a self-similar tiling of the line, or as a broken line in a d-dimensional space (where d is the number of letters, whose projection on a well-chosen line gives this self similar tiling. We will show how this construction can be formalised in a very natural way, so that the substitutions acts on the set Z^d of integer points, on the set of broken lines, and more generally, on the set of k-dimensional faces for any 0≤k≤d. If the substitution is unimodular, the dual maps are also defined, and act as a kind on inverse to the substitution. In this way, we can extend a number of theorems known in the Pisot case to the general hyperbolic case. We will show some examples.

Ion Wood-Thanan (Cardiff University, UK)

Title: A quantum mechanical road-coloring theorem

Abstract: The road-coloring theorem is a statement that, for certain type of directed graphs, it is always possible to color the graph as to permit a synchronizing instruction. A synchronizing instruction allows one to start at any vertex on the graph and travel to the same final vertex. Inspired by this theorem, we attempted to create quantum tight-binding models – models which are naturally represented on a graph – endowed with a synchronizing instruction. In the context of mechanics, a synchronizing instruction allows one to evolve a set of initial states to the same final states using the same time-dependent potential. The required irreversible time-evolution necessitates that the quantum system has a non-Hermitian Hamiltonian; the system is open and can have gain and loss from the environment. The talk will be centered around two attempts at creating these quantum road-colored systems: one attempt based on ancilla qubits used in quantum computing and one attempt based on the Hatano-Nelson model, a foundational non-Hermitian tight-binding model. 

Coffee Break

Gabriel Fuhrmann (Durham University, UK)

Title:  On the lack of equidistribution on fat Cantor sets

Abstract:  Given an irrational rotation, it is straightforward to see that for every Cantor set C, there is a dense (in fact, residual) set of points whose orbit does not intersect C. On the other hand, if C is a fat Cantor set (that is, of positive Lebesgue measure), almost every point visits C with a frequency equal to the measure of C. But what other frequencies of visits to C may occur? In the words of a recent MathOverflow post [1], what is the Birkhoff spectrum of fat Cantor sets?  We give a first answer to this question by showing that every irrational rotation allows for certain fat Cantor sets C whose Birkhoff spectrum is maximal, that is, equal to the interval [0,Leb(C)]. In this talk, we will focus on discussing some of the basic tools behind this result and extensions of it.

[1] D. Kwietniak, Possible Birkhoff spectra for irrational rotations, MathOverflow (2020), https://mathoverflow.net/q/355860 (version: 2020-03-27).

Grimm Network Meeting Dinner

Dishoom (One Chamberlain Square, Birmingham, B3 3AX)

We have two tables booked for the Grimm Network Meeting Dinner: one at 17:30 for those who need to catch a early evening train, and one at 18:00 for those living locally or staying overnight.