  In this paper we characterize tame dynamical systems and functions in terms
of eventual non-sensitivity and eventual fragmentability. As a notable
application we obtain a neat characterization of tame subshifts $X \subset
\{0,1\}^{\mathbb Z}$: for every infinite subset $L \subseteq {\mathbb Z}$ there
exists an infinite subset $K \subseteq L$ such that $\pi_{K}(X)$ is a countable
subset of $\{0,1\}^K$. The notion of eventual fragmentability is one of the
properties we encounter which indicate some "smallness" of a family. We
investigate a "smallness hierarchy" for families of continuous functions on
compact dynamical systems, and link the existence of a "small" family which
separates points of a dynamical system $(G,X)$ to the representability of $X$
on "good" Banach spaces. For example, for metric dynamical systems the property
of admitting a separating family which is eventually fragmented is equivalent
to being tame. We give some sufficient conditions for coding functions to be
tame and, among other applications, show that certain multidimensional
analogues of Sturmian sequences are tame. We also show that linearly ordered
dynamical systems are tame and discuss examples where some universal dynamical
systems associated with certain Polish groups are tame.

Dynamical Systems;Functional Analysis;General Topology