Description. The notion of chaos is ubiquitous; not only does it appear all over the place, but also there exist many different definitions of what it is to be considered chaotic.

In this project, we will study chaos in one-dimensional (topological) dynamics. This is dynamics from more of a ‘pure’ maths perspective. A one-dimensional topological dynamical system is nothing but a continuous self-map T : I → I where I denotes either the unit interval or the circle and T is to be understood as a discrete time evolution (i.e., time is modelled by the natural numbers). The goal of the theory of dynamical systems is to develop mathematical tools to understand how points x ∈ I move as we iteratively apply the map T to x. Roughly speaking, if this motion is seemingly random or apparently unpredictable one calls T chaotic.

It turns out that there are many ways to formalise this intuitive idea. After laying the foundations for a general understanding of dynamical systems theory, this project could study some of the most important notions of chaos such as positive topological entropy, Devaney Chaos, mixing, or Li-Yorke Chaos.

Prerequisites. This project is mainly topological in nature and a general background of level 2 pure maths, including specifically a little metric spaces theory, is all that is needed. It would be helpful to take Topology III and Analysis III as co-requisites.

References. It is good to use a variety of sources, but here are two to start having a look at. [1] is a pretty comprehensive reference on one-dimensional dynamics but at times certainly quite hard to read if you are just entering the topic. It may, nonetheless, be useful depending on the direction chosen in the project. It can be downloaded from the author’s website here. [2] provides a smooth and well-written introduction to the field and fits perfectly to this project. You can download it here.

References