  The well-known ergodic hierarchy of sheerly ergodic, mixing, Kolmogorov and
Bernoulli systems, with each next level supposedly encompassing the previous
one, is shown to be too simplistic in its usual formulation. A K-system can be
sheerly ergodic and sometimes may be reduced to a sheerly mixing system by some
simple projection. More precise characterizations of ergodic properties of
dynamical systems should start out from a consideration of the full Lyapunov
spectrum.
