On some common misconceptions regarding the "Ergodic Hierarchy"

TL;DR

This paper challenges simplified views of the ergodic hierarchy, emphasizing the need for more precise characterizations based on the full Lyapunov spectrum to better understand dynamical systems.

Contribution

It clarifies misconceptions about the ergodic hierarchy and proposes that detailed spectral analysis is essential for accurate classification.

Findings

K-systems can be purely ergodic and reducible to mixing systems

Simplistic hierarchical assumptions are inadequate for complex systems

Lyapunov spectrum provides a more precise ergodic property characterization

Abstract

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.