Tame dynamical systems

Tame dynamical systems were introduced by A. K\"{o}hler in 1995 and their theory was 
developed during last decade in a series of works by several authors. Connections to 
other areas of mathematics like: Banach spaces, model theory, tilings, cut and project 
schemes were established. A metric dynamical $G$-system $X$ is tame if every element 
$p \in E(X)$ of the enveloping semigroup $E(X)$ is a limit of a sequence of elements 
from $G$. In a recent joint work with Eli Glasner we study the following general question:
which finite coloring $G \to \{0, \dots ,d\}$ of a discrete countable group $G$ defines a 
tame minimal symbolic system $X \subset \{0, \dots ,d\}^G$. Any Sturmian bisequence 
$\Z \to \{0,1\}$ on the integers is an important prototype.
As closely related directions we study cutting coding functions coming from circularly ordered 
systems. As well as generalized Helly's sequential compactness type theorems about families 
with bounded total variation. We show that circularly ordered dynamical systems are tame and 
that several Sturmian like symbolic  $G$-systems are circularly ordered.