Published online by Cambridge University Press: 04 February 2021
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.