  Under a natural assumption on the dynamical degrees, we prove that the Green
currents associated to any H\'enon-like map in any dimension have H\"older
continuous super-potentials, i.e., give H\"older continuous linear functionals
on suitable spaces of forms and currents. As a consequence, the unique measure
of maximal entropy is the Monge-Amp\`ere of a H\"older continuous
plurisubharmonic function and has strictly positive Hausdorff dimension. Under
the same assumptions, we also prove that the Green currents are woven. When
they are of bidegree $(1,1)$, they are laminar. In particular, our results
generalize results known until now only in algebraic settings, or in dimension
2.
