Maps close to identity and universal maps in the Newhouse domain

TL;DR

This paper demonstrates that in the Newhouse domain, generic two-dimensional maps are C^r-universal, meaning their iterations can approximate any two-dimensional diffeomorphism, revealing complex dynamical behaviors.

Contribution

It establishes that C^r-close to identity maps form a residual set of renormalized iterations, and shows universality of certain maps in the Newhouse domain.

Findings

Generic maps in the Newhouse domain are C^r-universal.

Renormalized iterations of near-identity maps form a residual set.

Universal maps exhibit infinitely many coexisting hyperbolic attractors and repellers.

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