Poincaré rotation numbers and Riesz and Voronoi means.
type: Item
instance of: scholarly article
title: Poincaré rotation numbers and Riesz and Voronoi means.
zbMATH Open document ID: 1073.37049
DOI: 10.1023/A:1025020611681
author: Valery V. Kozlov
author: Tatiana Kozlova
published in: Mathematical Notes
publication date: 2004-12-02T00:00:00Z
review text: We say that a sequence of numbers (not necessarily converging) converges in Cesàro sense, if their averages converge. We say that it converges in the sense of Riesz with given weights \(p_0, p_1,\dots\), \(p_0>0\), \(p_i\geq0\), if their averages with the weights \(p_i\) converge. We say that a sequence converges in the sense of Voronoi with given weights as above, if the latter convergence statement holds true, but for averages taken with the inverse order of weights. Given a circle homeomorphism \(T:S^1\to S^1= \mathbb R\slash2\pi\mathbb Z\), \(T(x)=x+f(x)\). Poincaré's theorem implies that for any \(x\) the sequence \(T^n(x)\) converges to the rotation number of \(T\) in the sense of Cesàro. The paper under review gives sufficient conditions on a weight sequence \(p_i\) for convergence of a sequence \(T^n(x)\) (with arbitrary \(T\) and \(x\)) in either Riesz or Voronoi sense. The first result (Theorem 1) affirms Voronoi convergence of \(T^n(x)\) , whenever the weight sequence \(p_i\) is nonincreasing and forms a diverging series. The second result (Theorem 2) affirms Riesz convergence of \(T^n(x)\), whenever the weights \(p_i\) are nondecreasing and in addition \(\frac{p_n} {\sum_0^np_s}\to0\). In fact, Theorem 1 and 2 are equivalent. The authors also give the following weaker sufficient condition for Riesz convergence in Theorem 2: \(\frac{p_n} {\sum_0^np_s}\to0\), and there exists an \(H>0\) such that for any \(n\) one has \[ p_0(n+1)+| p_1-p_0| n+\dots+ | p_n-p_{n-1}| \leq H(p_0+\dots+p_n). \] Theorem 2 follows from Theorem 3. In the proof of Theorem 3 the authors use results from [\textit{G. H. Hardy}, Divergent series, Oxford: At the Clarendon Press (Geoffrey Cumberlege) XIV (1949; Zbl 0032.05801)] on relation between Voronoi and Cesàro convergences.
Mathematics Subject Classification ID: 37E45
Mathematics Subject Classification ID: 37E10
zbMATH DE Number: 2121039
zbMATH Keywords: circle homeomorphism
zbMATH Keywords: rotation number
zbMATH Keywords: Cesàro convergence
zbMATH Keywords: Riesz convergence
zbMATH Keywords: Voronoi convergence
reviewed by: Alexey A. Glutsyuk
MaRDI profile type: Publication
Recommended article: Rotation numbers, eigenvalues, and the Poincaré--Birkhoff theorem
Recommended article: Rigorous enclosures of rotation numbers by interval methods
Recommended article: A Note on the Rotation Number of Poincare
Recommended article: Empirical means for the Voronoi tessellations of the Poincaré disc.
Recommended article: Rotation numbers for measure-valued circle maps
Recommended article: Q5692697
Recommended article: Rotation Distance, Triangulations, and Hyperbolic Geometry
description: scientific article; zbMATH DE number 2121039