Abstract: Motivated by a problem in the diffraction of certain 'bijective' constant length substitutions, one is naturally lead to a generalisation of the problem to substitutions on a circular alphabet, meaning we have to leave the realm of finite alphabets. I will give a brief introduction to the motivating problem and how this generalisation helps us to solve it in the special case of so-called `abelian' substitutions. We can fully characterise the spectral type of their diffraction and exactly when they are periodic.

The majority of the talk though will be a tour through the necessary theory developed for substitutions on compact Hausdorff alphabets and the corresponding topological dynamics associated with their subshifts. There are still lots of open questions, and so a ground-level introduction to these systems will hopefully be approachable and stimulating. This is joint work with Neil Manibo and Jamie Walton.