2010/09/04 by Dmitry Turaev, Turaev, Dmitry · 1 citation
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1009.0858
openalex publication_date 2010/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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