Maps close to identity and universal maps in the Newhouse domain

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Given an n-dimensional Cʳ-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some Cʳ-coordinates in which the ball acquires radius 1. We show that for any r >/- 1 the renormalized iterations of Cʳ -close to identity maps of an n-dimensional unit ball Bⁿ (n >/- 2) form a residual set among all orientation-preserving Cʳ -diffeomorphisms Bⁿ → Rⁿ. In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of Cʳ -close to identity maps, with the same dimension of the phase space. As an application, we show that any Cʳ-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 Cʳ -universal in the sense that its iterations, after an appropriate coordinate transformation, Cʳ -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