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- W4297447773 abstract "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" @default.
- W4297447773 created "2022-09-28" @default.
- W4297447773 creator A5084996461 @default.
- W4297447773 date "2010-09-04" @default.
- W4297447773 modified "2023-09-29" @default.
- W4297447773 title "Maps close to identity and universal maps in the Newhouse domain" @default.
- W4297447773 doi "https://doi.org/10.48550/arxiv.1009.0858" @default.
- W4297447773 hasPublicationYear "2010" @default.
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