Invariant measure for the trigonometric thermostat

Determine whether the trigonometric Nosé–Hoover oscillator on the three-dimensional torus possesses a physically meaningful invariant measure, thereby characterizing its statistical properties.

Background

The trigonometric Nosé–Hoover oscillator is a periodic dynamical system defined on the compact phase space $\mathbb T^3$. Although the paper analyzes its regular, chaotic, and phase-space-contracting regimes, it does not establish the existence or physical relevance of an invariant statistical measure analogous to the canonical measure associated with the classical Nosé–Hoover system.

The authors identify the statistical behavior of the thermostat as an explicit open problem because numerical evidence of chaotic attracting sets and nonuniform contraction does not by itself determine whether a physically meaningful invariant measure exists.