  We consider compact invariant sets \Lambda for C^{1} maps in arbitrary
dimension. We prove that if \Lambda contains no critical points then there
exists an invariant probability measure with a Lyapunov exponent \lambda which
is the minimum of all Lyapunov exponents for all invariant measures supported
on \Lambda. We apply this result to prove that \Lambda is uniformly expanding
if every invariant probability measure supported on \Lambda is hyperbolic
repelling. This generalizes a well known theorem of Mane' to the
higher-dimensional setting.
