Description. A (topological) dynamical system is essentially a self-homeomorphism ϕ : X → X on a compact metrizable space X. The overall goal of the theory of dynamical systems is to develop mathematical tools to understand how points x ∈ X move as we iteratively apply the map ϕ to x. This may occur in a very unpredictable fashion (in the presence of chaos) or in a very orderly fashion (in particular, for so-called equicontinuous systems).

A symbolic dynamical system is a dynamical system where X is the Cantor set or, equivalently, the set of bi-infinite 0–1-sequences (equipped with an appropriate metric). (In formal terms, X = Σ := {0,1}^ℤ.) In some sense, one can say that understanding all symbolic dynamical systems amounts to understanding every topological dynamical system. (If you want to convince a non-mathematician of the richness of symbolic dynamics, you may argue as follows: every computation any computer could possibly carry out is nothing but the successive application of one and the same rule [defined by the architecture of the computer] to a finite 0–1-sequence [given by the current state of the computer, that is, the data in its memory and your input].)

An important sub-class of symbolic dynamical systems (and in a sense, the building blocks of symbolic dynamical systems) are minimal symbolic systems. Already the class of minimal symbolic systems is too rich to ever be fully understood but it contains certain families (such as Sturmian subshifts, Toeplitz shifts or adding machines) where, due to intensive research efforts, much is known.

In this project, you would first of all study the foundations of topological and symbolic dynamical systems (unless you are already familiar with that) to then move further to more specific minimal symbolic systems and specialise in one of the infinitely many topics.

Pre-/Co-requisites. A good understanding of basic concepts from topology (in particular compactness and continuity) on the level of Topology 3 or Analysis 3. Desirable but not necessary would be some background in measure theory (again, on the level of Analysis 3). Similarly, it will certainly help (but is again not necessary) to take Ergodic Theory and Dynamics IV.

References. For a general introduction to dynamical systems (and some symbolic dynamics), you may take a look at [1] (available as an online copy at our library). Closer to the specific topic of this project but also with a general introduction to topological dynamics is [2] (an online copy can be found on the author’s website http://www.cts.cuni.cz/~kurka/studij.html). There are plenty of excellent other online resources, so consider the provided references only as some incredibly incomplete suggestion for a first read.