The Liouville property of invariant measures of completely integrable systems and the Monge-Ampère equation
type: Item
instance of: scholarly article
title: The Liouville property of invariant measures of completely integrable systems and the Monge-Ampère equation
zbMATH Open document ID: 0803.58033
DOI: 10.1007/BF01210221
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: Let \(D\) be a domain in \(\mathbb{R}^ n\), \(I = (I_ 1,\dots,I_ n) \in D\), \(\varphi = (\varphi_ 1,\dots, \varphi_ n)\) be a set of angle coordinates which parametrizes the \(n\)-dimensional torus \(T^ n\) and let \(\omega_ 1,\dots, \omega_ n\) be functions of \(I\). Consider the non- degenerate case \(\partial(\omega_ i)/\partial(I_ i) \neq 0\) of the motion of the system \(\dot I_ 1 = \dots =\dot I_ n = 0\), \(\dot \varphi_ 1 = \omega_ 1,\dots, \dot \varphi_ n = \omega_ n\) on the \(n\)-dimensional tori \(I = \text{const}\), which is quasi-periodic with frequencies \(\omega_ 1,\dots, \omega_ n\). The main result of this paper is that any invariant measure of the completely integrable system defined above which has a continuous positive density is Liouville. The proof of this result reduces to the solvability of a higher-dimensional analogue of the Monge-Ampère equation.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 567915
zbMATH Keywords: Liouville property
zbMATH Keywords: completely integrable systems
zbMATH Keywords: invariant measure
zbMATH Keywords: Monge-Ampère equation
author: Valery V. Kozlov
reviewed by: Ubiratan D'Ambrosio
MaRDI profile type: Publication
cites work: Q5587708
cites work: Q3712656
cites work: Q3221462
cites work: Q5822604
full work available at URL: https://doi.org/10.1007/bf01210221
OpenAlex ID: W2069004508
description: scientific article; zbMATH DE number 567915