  We introduce a class of infinitely renormalizable, unicritical
diffeomorphisms of the disk (with a non-degenerate "critical point"). In this
class of dynamical systems, we show that under renormalization, maps eventually
become H\'enon-like, and then converge super-exponentially fast to the space of
one-dimensional unimodal maps. We also completely characterize the local
geometry of every stable and center manifolds that exist in these systems. The
theory is based upon a quantitative reformulation of the Oseledets-Pesin theory
yielding a unicritical structure of the maps in question comprising regular
Pesin boxes co-existing with "critical tunnels" and "valuable crescents". In
forthcoming notes we will show that infinitely renormalizable perturbative
H\'enon-like maps of bounded type belong to our class.
