  We develop non-invertible Pesin theory for a new class of maps called cusp
maps. These maps may have unbounded derivative, but nevertheless verify a
property analogous to $C^{1+\epsilon}$. We do not require the critical points
to verify a non-flatness condition, so the results are applicable to
$C^{1+\epsilon}$ maps with flat critical points. If the critical points are too
flat, then no absolutely continuous invariant probability measure can exist.
This generalises a result of Benedicks and Misiurewicz.
