  Given an n-dimensional C^r-diffeomorphism g, its renormalized iteration is an
iteration of g, restricted to a certain n-dimensional ball and taken in some
C^r-coordinates in which the ball acquires radius 1. We show that for any r >/-
1 the renormalized iterations of C^r -close to identity maps of an
n-dimensional unit ball B^n (n >/- 2) form a residual set among all
orientation-preserving C^r -diffeomorphisms B^n \to R^n. In other words, any
generic n-dimensional dynamical phenomenon can be obtained by iterations of C^r
-close to identity maps, with the same dimension of the phase space. As an
application, we show that any C^r-generic two-dimensional map which belongs to
the Newhouse domain (i.e., it has a wild hyperbolic set, so it is not
uniformly-hyperbolic, nor uniformly partially-hyperbolic) and which neither
contracts, nor expands areas, is C^r -universal in the sense that its
iterations, after an appropriate coordinate transformation, C^r -approximate
every orientation-preserving two-dimensional diffeomorphism arbitrarily well.
In particular, every such universal map has an infinite set of coexisting
hyperbolic attractors and repellers
