Maps Close to Identity and Universal Maps in the Newhouse Domain

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 a parts per thousand yen 1 the renormalized iterations of C (r) -close to identity maps of an n-dimensional unit ball B (n) (n a parts per thousand yen 2) form a residual set among all orientation-preserving C (r) -diffeomorphisms B (n) -> 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 that belongs to the Newhouse domain (i.e., it has a so-called wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and that 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.