Let $I=[0,1]$, and let $\mathcal J$ be the class of functions $I \rightarrow I$
with connected $G_{\delta}$ graph. Recently it was shown that dynamical
systems generated by maps in $\mathcal J$ have some nice properties. Thus,
the Sharkovsky's theorem is true, and a map has positive topological entropy
if and only if every periodic point has period $2^{n}$, for an integer $n\ge 0$.
In this paper we consider, for a map $\varphi$ in $\mathcal J$, properties
of minimal sets, i.e., sets $M\subset I$ such that the $\omega$-limit set
$\omega_{\varphi }(x)$ is $M$, for every $x \in M$. If $\varphi $
is continuous then, as is well-known, $M$ is minimal if and only if
$M$ is non-empty, closed, $\varphi (M)\subseteq M$, any point in $M$
is uniformly recurrent, and no proper subset of $M$ has these properties.
In this paper we prove that the same is true for $\varphi\in\mathcal J$ with zero
topological entropy, but not for an arbitrary $\varphi\in\mathcal J$.
We also show relations between other properties which are equivalent
in the class of continuous maps of the interval.