Changing rotation intervals of endomorphisms of the circle
type: Item
instance of: scholarly article
title: Changing rotation intervals of endomorphisms of the circle
zbMATH Open document ID: 0603.58036
DOI: 10.1007/BF01388962
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143311
review text: Let \(E^ r_ 1(S^ 1)\) be the set of endomorphisms of the circle \(S^ 1\), of degree one and class \(C^ r\) (r\(\geq 0)\), endowed with the \(C^ r\) topology. For any \(\tilde f\in E^ r_ 1(S^ 1)\), denote by f a lift of \(\tilde f\) to \({\mathbb{R}}\). Then, following Newhouse, Palis and Takens, define the rotation set of \(\tilde f\) by \[ \rho (\tilde f)=\rho (f)=cl.\{\limsup_{n\to \infty}(f^ n(x)-x)/n| x\in {\mathbb{R}}\}. \] Denote by \(\Sigma^ r\) the subset of endomorphisms in \(E^ r_ 1(S^ 1)\) for which the rotation interval remains unchanged by \(C^ r\) small perturbations. The aim of this paper is to prove that \(\Sigma^ r\) is an open dense set in \(E^ r_ 1(S^ 1)\) for \(r\geq 1\), and to characterize the endomorphisms in \(\Sigma^ r\). The authors also characterize the endomorphisms in the boundary of a connected component of \(\Sigma^ r\).
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 3973907
zbMATH Keywords: rotation intervals
zbMATH Keywords: endomorphisms
zbMATH Keywords: \(C^ r\)-persistence
zbMATH Keywords: small perturbations
author: Rodrigo Bamón
author: Iaci Malta
author: M. J. Pacifico
reviewed by: Ding Tongren
MaRDI profile type: Publication
cites work: Rotation sets are closed
cites work: Q4746554
cites work: Bifurcations and stability of families of diffeomorphisms
description: scientific article; zbMATH DE number 3973907