  Aspects of the Nos\'e and Nos\'e-Hoover dynamics developed in 1983-1984 along
with Dettmann's closely related dynamics of 1996, are considered. We emphasize
paradoxes associated with Liouville's Theorem. Our account is pedagogical,
focused on the harmonic oscillator for simplicity, though exactly the same
ideas can be, and have been, applied to manybody systems. Nos\'e,
Nos\'e-Hoover, and Dettmann flows were all developed in order to access Gibbs'
canonical ensemble directly from molecular dynamics. Unlike Monte Carlo
algorithms dynamical flow models are often not ergodic and so can fail to
reproduce Gibbs' ensembles. Accordingly we include a discussion of ergodicity,
the visiting of all relevant microstates corresponding to the desired ensemble.
We consider Lyapunov instability too, the usual mechanism for phase-space
mixing. We show that thermostated harmonic oscillator dynamics can be
simultaneously expanding, incompressible, or contracting, depending upon the
chosen "phase space". The fractal nature of nonequilibrium flows is also
illustrated for two simple two-dimensional models, the hard-disk-based Galton
Board and the time-reversible Baker Map. The simultaneous treatment of flows as
one-dimensional and many-dimensional suggests some interesting topological
problems for future investigations.
