Derivatives of rotation number of one parameter families of circle diffeomorphisms
type: Item
instance of: scholarly article
title: Derivatives of rotation number of one parameter families of circle diffeomorphisms
zbMATH Open document ID: 1244.37026
author: Shigenori Matsumoto
published in: Kodai Mathematical Journal
publication date: 2012-05-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.2591
full work available at URL: https://projecteuclid.org/euclid.kmj/1333027257
reviewed by: Krzysztof Ciepliński
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 37E45
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 6038721
zbMATH Keywords: circle diffeomorphism
zbMATH Keywords: rotation number
zbMATH Keywords: Liouville number
zbMATH Keywords: Denjoy distortion lemma
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: invariant measure
MaRDI profile type: Publication
OpenAlex ID: W1982638492
arXiv ID: 1103.2591
DOI: 10.2996/KMJ/1333027257
review text: For \(t\in [-\frac{1}{2}, \frac{1}{2}]\) and an orientation-preserving \(C^{\infty }\)-diffeomorphism \(f\) of the circle set \(f_t :=R_t \circ f\), where \(R_t\) is the rotation by \(t\). The author shows that if the rotation number \(\rho (t)\) of \(f_t\) is irrational, then NEWLINE\[NEWLINE \limsup _{s\to t} \frac{\rho (s)-\rho (t)}{s-t}\geq 1. NEWLINE\]NEWLINE (He also remarks that this inequality is not valid for \(\liminf\).) To this end, a weaker version of the above result is first proved. (This version allows the author to give a new proof of a result of \textit{J. Graczyk} in [Bol. Soc. Bras. Mat., Nova Sér. 24, No. 2, 201--210 (1993; Zbl 0798.58048)].) Next, with the use of the Denjoy distortion lemma, the case when \(\rho (t)\) is a Liouville number is considered. Finally, the case of non-Liouville \(\rho (t)\) is studied. To do this, a result of \textit{P. Brunovský} from [Czech. Math. J. 24(99), 74--90 (1974; Zbl 0308.58007)] is applied.
description: scientific article; zbMATH DE number 6038721