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Abstract. The Nosé-Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton's equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé-Hoover dynamics being ergodic. Hoover presented numerical experiments that show the Nosé-Hoover dynamics to be non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé-Hoover thermostat does not give an ergodic dynamics for the one-dimensional harmonic oscillator when the "mass" of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé-Hoover dynamical system that separate phase space into invariant regions.We present numerical experiments motivated by our analysis that seem to show that the dynamics is not ergodic even for a moderate thermostat mass. We also give numerical experiments of the Nosé-Hoover chain with two thermostats applied to the one-dimensional harmonic oscillator. These experiments seem to support the non-ergodicity of the dynamics if the masses of the reservoirs are large enough and are consistent with ergodicity for more moderate masses.

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Abstract. Central configurations are important special solutions of the Newtonian N-body problem of celestial mechanics. In this paper a highly symmetrical case is studied. As the masses are varied, spatial central configurations appear through bifurcation from planar ones. In particular, spatial configurations can be found which are arbitrarily close to being planar.