On some common misconceptions regarding the "Ergodic Hierarchy"

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.