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One of the strongest regularity properties an aperiodic pattern can have is of being linearly repetitive, which means that finite sub-patterns are spaced in a highly regular manner across the pattern leaving only small gaps between them. Substitution tilings (under standard restrictions) always have linear repetitivity but the question for cut and project sets is more difficult. In this talk I will give a simple introduction to both the cut and project method and the property of linear repetitivity. After this I will explain recent joint results with Henna Koivusalo on characterising linear repetitivity for cut and project sets whose windows are convex polytopes, which connects the dynamics of aperiodic patterns with the branch of Number Theory called Diophantine Approximation.
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A solenoidal manifold is an example of a foliated bundle with Cantor set fibre and foliation by path-connected components. The dynamics of a solenoidal manifold is given by the monodromy action of the fundamental group of the base space on the fibre, and, depending on the group, it may exhibit interesting dynamical phenomena.
In a series of papers, joint with Steve Hurder, I developed invariants which classify solenoidal manifolds up to return equivalence. In this talk I will discuss a new invariant, recently introduced in a joint paper with Hurder, called the prime spectrum of a solenoidal manifold. I will show that, when the acting group is nilpotent, if the prime spectrum consists of a finite number of primes, then the monodromy action is always stable, while the actions with infinite prime spectra may be wild. This result has connections with the results about self-embeddings of groups, joint with Hurder and van Limbeek. The notion of a stable action for pseudogroups on topological spaces was introduced by Alvarez Lopez and Candel, who called such actions quasi-analytic or quasi-effective.
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Suppose \(P\) is some decoration of \(d\)-dimensional Euclidean space \(E\), that is, a structure such as a tiling of, or Delone set in, \(E\). We consider the images of \(P\) under all rigid motions of \(E\). Under mild assumptions, and possibly after suitable completion, these form the points of a compact space with the structure of a \(d+(d-1)/2\) dimensional matchbox manifold. In this talk we introduce this this space and are particularly interested - though not exclusively - in the case that \(P\) is aperiodic. Tools from homotopy and shape theory give a way of analysing \(P\); in the special case of a periodic tessellation of \(E\) this recovers the classical space group, and in the case of \(P\) a model set for a quasicrystal, the crystallographers’ “aperiodic space groups” appear, but the construction is general. This is joint work with Jamie Walton.
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