  Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young
introduced entropies along unstable foliations for an ergodic measure $\mu$. We
relate those entropies to covering numbers in order to give a new upper bound
on the metric entropy of $\mu$ in terms of Lyapunov exponents and topological
entropy or volume growth of sub-manifolds. We also discuss extensions to the
$C^{1+\alpha},\,\alpha>0$ case.
