Symbolic Dynamics

A (discrete) dynamical system consists of a compact metric space and a continuous map $\varphi : X \to X$ . In this course we study the important examples of such a dynamical systems known as subshifts of finite type or Markov shifts. To describe these consider a finite set $F = \{1,2,\cdots n\}$ . Let X consist of the sequences

of elements from F. X is a compact metric space, e.g. in the metric

$\begin{displaymath}D(x,y) = \sum_{i=1}^{\infty} 2^{-i}\vert x_i - y_i\vert . \end{displaymath}$

Define a continuous map $\sigma$ ('the shift') on X by

$\begin{displaymath}\sigma(x)_i = x_{i+1}. \end{displaymath}$

Then $(X, \sigma)$ is a dynamical system. More generally we shall consider a closed $\sigma$ -invariant subset X_A determined by an $n \times n$ -matrix A = (A_ij) with entries from $\{0,1\}$ in the following way:

$\begin{displaymath}X_A = \{ x \in X : \ A_{x_ix_{i+1}} = 1 \ \forall i \} . \end{displaymath}$

Since $\sigma(X_A) \subseteq X_A$ , $(X_A,\sigma)$ is a dynamical system known as a one-sided subshift of finite type. In the course we shall study such systems and in particular obtain necessary and sufficient conditions for two one-sided subshifts of finite type to be conjugate. (Two dynamical systems $(X,\varphi)$ and $(Y,\psi)$ are conjugate when there is a homeomorphism $h : X \to Y$ such that $\psi \circ h = h \circ \varphi$ .)
We will also consider two-sided subshifts of finite type which are defined similarly from bi-infinite sequence $\{x_i\}_{i \in \Bbb Z}$ and address the same questions for these.

Qualifications
Geometry 1, Analysis 1.

Textbook
Bruce P. Kitchens: Symbolic Dynamics, Springer Verlag, Berlin 1997.

Revideret 24.11.2022

Studievejledningen ved Faculty of Natural Sciences og Faculty of Technical Sciences