Unique ergodicity of dynamic systems on the torus
type: Item
instance of: scholarly article
title: Unique ergodicity of dynamic systems on the torus
zbMATH Open document ID: 0871.58058
author: Jun Le
author: Zuoling Zhou
published in: Chinese Science Bulletin
publication date: 1997-10-06T00:00:00Z
review text: The authors consider a dynamical system on the torus \(T^2\) defined by \[ \frac{dx}{dt}=G(x,y),\quad \frac{dy}{dt}=H(x,y), \] where \(G\) and \(H\) are continuous functions of period 1 in \(x\) and \(y\). In the region \(0\leq x<1\), \(0\leq y<1\), the only singular point is \(x=0\), \(y=0\). Let \(f\) denote the \(C^1\)-flow associated with this system. The authors prove that when the rotation number of \(f\) is either rational or infinite, then \(f\) is uniquely ergodic if and only if \(f\) has no periodic orbit. They also show that an \(f\)-invariant probability measure \(m\) on \(T^2\) is a nontrivial ergodic measure if and only if the rotation number of \(f\) is irrational and \(m(A)=0\) where \(A\) consists of the singular point and all other points whose \(\alpha\)- or \(\omega\)-limit set is the singular point. Finally, the authors demonstrate that \(f\) has at most one non-trivial ergodic measure.
reviewed by: William J. Satzer Jr.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 987495
zbMATH Keywords: unique ergodicity
zbMATH Keywords: dynamical system
zbMATH Keywords: rotation number
zbMATH Keywords: ergodic measure
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 987495