We consider six properties of continuous maps from compact interval into
itself: (i) $h(f)=0$; (ii) $\Rec (f)$ is $F_\sigma$; (iii) $f$ is Lyapunov stable
on $\Per(f)$; (iv) for every simple system $(I_{n})_{n \in \N}$ of intervals
associated with an infinite $\omega$-limit set, $\lim_{n\to\infty}\max_{0\leq i <
2^{n}} \lambda (f^{i} (I_{n})) = 0$; (v) $\Per (f)$ is $G_\delta$; (vi) every
linearly ordered chain of $\omega$-limit sets is countable. They were believed to
be equivalent. Recently some authors provide counterexamples showing that this is
not the case. In this paper we give all the relations between these properties.
We show that (iv) $\Rightarrow$ (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i), (iv)
$\Rightarrow$ (vi) $\Rightarrow$ (i), and (v) $\Rightarrow$ (i), and that there
is no other implication between these properties.