Maps close to identity and universal maps in the newhouse domain
File(s)
Accepted version
Author(s)
Turaev, Dmitry
Type
Journal Article
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.
Date Issued
2015-05-01
Date Acceptance
2014-12-30
Citation
Communications in Mathematical Physics, 2015, 335 (3), pp.1235-1277
ISSN
0010-3616
Publisher
Springer
Start Page
1235
End Page
1277
Journal / Book Title
Communications in Mathematical Physics
Volume
335
Issue
3
Copyright Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-015-2338-4
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000351224000008&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Physics, Mathematical
Physics
HOMOCLINIC TANGENCIES
GLOBAL BIFURCATIONS
DIFFEOMORPHISMS
HYPERBOLICITY
ATTRACTORS
ABUNDANCE
DYNAMICS
SYSTEMS
Publication Status
Published
Date Publish Online
2015-03-06