Description. Bratteli(-Vershik) diagrams originated in the theory of operator algebras in the 1970s. Somewhat surprisingly, they nowadays have interesting applications in a variety of fields, including ergodic theory, the theory of dynamical systems and the study of aperiodic tilings, where they provide a language for describing ideas from both topology and analysis. At the heart of these applications is the utility of Bratteli diagrams to model self maps of the Cantor set and their associated dynamics (the long term behaviour of repeatedly applying such a self map). The first step of this project would be to focus on the relation between dynamics and Bratteli diagrams and then turn to applications.

In brief, a Bratteli diagram is an infinite graph B = (V,E) where the vertex set V = ⋃ _i≥0V _i (with V ₀ a singleton) and the edge set E = ⋃ _i≥0E_i are partitioned into disjoint finite subsets V _i and E_i such that

• each edge in E_i starts at vertex in V _i and ends at some vertex in V _i+1;
• each vertex in V _i is connected (through some e ∈ E_i) to a vertex in V _i+1 and vice versa.

The key idea is to focus on the set, in fact in a natural way a topological space, X of paths from V ₀ to infinity; it turns out this can be identified with a Cantor set. A Bratteli-Vershik diagram is a Bratteli diagram endowed with a certain kind of order ω on X. The successor map induced by ω defines a homeomorphism on X.

It turns out that every self-homeomorphism on a Cantor set can be described by a suitably chosen Bratteli-Vershik diagram. This should be the first result to aim at. Alongside, one might usefully study the basics of dynamics and ergodic theory as well as the theory of subshifts – debatably the most important class of dynamical systems on the Cantor set. Once the machinery is set up, it can be applied in a variety of ways, for example, to prove that every Choquet simplex is realised as the simplex of invariant measures associated to certain minimal subshifts, or to use it as a tool to compute cohomological or K-theoretic information.

Prerequisites. A good understanding of basic concepts from topology (in particular compactness and continuity) suffices, and some understanding of measure theory may be useful. The module Topology III would more than cover the first of these, Analysis III (or Analysis IV studied alongside) would contain the measure theory one might want. Algebraic Topology IV might be of interest if one wanted to follow the cohomological and K-theory applications.

References. For a very nice and general introduction to dynamical systems and ergodic theory, [1] is a good read. A nice exposition of the relation between (minimal) dynamical systems on the Cantor set and Bratteli-Vershik diagrams can be found in [2] (for a start, we’d suggest reading Section 2–3 and possibly the first two pages of Section 1). See also [3]. Further, [4] not only contains the proof of the above-mentioned statement concerning the realisation of Choquet simplices but also comes with a well-written introductory section which is a good starting point for the project. All references are available at the library; with the exception of [2], you can find online copies there.

References