The Liouville property of invariant measures of completely integrable systems and the Monge-Ampère equation
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.
scientific article; zbMATH DE number 567915