  We study Henon maps which are perturbations of a hyperbolic polynomial p with
connected Julia set. We give a complete description of the critical locus of
these maps. In particular, we show that for each critical point c of p, there
is a primary component of the critical locus asymptotic to the line y = c.
Moreover, primary components are conformally equivalent to the punctured disk,
and their orbits cover the whole critical set. We also describe the holonomy
maps from such a component to itself along the leaves of two natural
foliations. Finally, we show that a quadratic Henon map taken along with the
natural pair of foliations, is a rigid object, in the sense that a conjugacy
between two such maps respecting the foliations is a holomorphic or
antiholomorphic affine map.
