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Week 2: DynamicsDynamical, arithmetical and algebraic aspects (June 6 - 10)
Mind the boatDeparture from La Rochelle sunday (june 5) at 7.30pm, no boat on monday! Schedule
Afternoons are devoted to talks and free discussions.You can already propose a talk by email. LecturesVan Cyr: Zero entropy symbolic dynamical systems Slides1 Slides2Symbolic dynamics is the study of the left-shift map σ acting on various closed subsets of the space AZ of all A-colorings of Z (where A is a finite alphabet). At first blush these systems seem simple, but looks can be deceiving: the famous Jewett- Krieger theorem states that any ergodic, probability-preserving dynamical system (X, T, B, μ) with finite entropy is (measure-theoretically) isomorphic to a subshift. Even as purely topological systems, subshifts provide examples that exhibit much of the richness of general systems while also being explicit enough for important problems to be tractable. In these lectures, I will survey some of the main theorems and open problems about symbolic dynamics with a focus on the study of systems with zero topological entropy. The first lecture will introduce the audience to the group of automorphisms of a symbolic system in one (or more) dimensions. The second lecture will survey some recent advances in the study of automorphisms of one-dimensional systems of zero entropy. The third and fourth lectures will study the ergodic properties of low complexity subshifts. Rostislav Grigorchuk: Groups of intermediate growth and aperiodic order Slides1 Slides2 Slides3 Slides4This series of lecture will start with a short introduction to growth of finitely generated groups and other algebraic and combinatorial objects (eg. graphs). It will continu with the construction of groups of intermediate growth (between polynomial and exponential). This will involve actions on rooted trees, finite Mealy type automata, dynamics on the boundary of the tree, Schreier graphs, and some other instruments. The next step will be to show how substitutions, and automatically generated sequences may appear in study of groups. This will be done via the notions of L-presentation and topological full group. Finally we will explain how groups of intermediate growth and amenable groups fits in all this panorama. Alan Haynes: Patterns, complexity, and repetitivity in cut and project sets Slides1 Slides2 Slides3 NotesWe will begin with classical results about word complexity, the Morse-Hedlund Theorem, and Sturmian sequences, and we will discuss the problem of classifying the collection of linearly repetitive words on a two letter alphabet. Next we will introduce cut and project sets, with examples and basic theory, and we will formulate the analogous higher dimensional problems of understanding pattern complexity and repetitivity. One of the goals of these lectures is to explain how geometric and dynamical viewpoints can be combined, together with input from Diophantine approximation, to provide a characterization of linearly repetitive cut and project sets which is a natural extension of the classical one-dimensional results. Daniel Lenz: Geometric and dynamical features of quasicrystals SlidesDelone sets with additional features are common models for quasicrystals. The additional features include finite local complexity, repetitivity and uniform patch frequencency. Taking the hull of a Delone set with finite local complexity, one obtains a dynamical system which can be seen as a geometric analogue of a subshift over a finite alphabet. Minimality and unique ergodicity of such systems are then characterized by repetitivity and uniform patch frequency respectively. A special role is played by linearly repetitive and densely repetive systems. We first give an introduction into the corresponding basic theory. We then turn to repetitive Meyer sets and discuss a bulk of work culminating recently in providing a hierarchy on how the associated dynamical systems map into their maximal equicontinuous factor. Finally, we have a look at diffraction and describe it via dynamical systems. Afternoon talks
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