## 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}→ 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.

Original language | English |
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Pages (from-to) | 1235-1277 |

Number of pages | 43 |

Journal | Communications in Mathematical Physics |

Volume | 335 |

Issue number | 3 |

DOIs | |

State | Published - 1 May 2015 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics