the hierarchy of mixing properties
Measure-preserving systems do not divide neatly into "ordered" and "random"; they form a graded ladder of ever-stronger forgetting properties. The hierarchy of mixing organises these levels so one can say precisely how chaotic a system is, and each rung is strictly stronger than the one below — there are examples separating every pair.
From weakest to strongest, the core ladder is: ergodic (time averages equal space averages; invariant sets are trivial) implies-by-weak mixing (continuous spectrum; correlations decay along density-one times; T x T ergodic) implies-by mixing (also written strong or 2-mixing: P(T^(-n) A intersect B) -> P(A) P(B) for all pairs) implies-by mixing of all orders (k-mixing: joint correlations of any k sets factorise as the time gaps grow) implies-by being a Kolmogorov (K-) system (trivial Pinsker/tail sigma-algebra; positive entropy on every nontrivial factor) implies-by Bernoulli (isomorphic to an iid shift — the gold standard of randomness, the top rung classified by Ornstein's theorem via entropy). The reverse implications all fail.
Why the ladder is useful: it converts the vague word "chaotic" into a checklist of testable, increasingly strong statements, and it tells you which limit theorems are available. Ergodicity buys a law of large numbers; mixing with rates buys central limit theorems; being Bernoulli says the system is, up to relabelling, literal coin-flips. A few honest signposts on the climb: the irrational rotation sits at the bottom (ergodic, not weakly mixing — it has discrete spectrum and zero entropy); horocycle flows are famously mixing of all orders but not Bernoulli-trivial to see; and Ornstein's deep theorem (entropy is a complete isomorphism invariant for Bernoulli shifts) is what makes the top of the ladder classifiable. Positive entropy enters only at the K-system rung; everything below can have zero entropy.
Placing four systems on the ladder: irrational rotation (ergodic only); a suitable [T, T^(-1)] or Chacon map (weakly mixing, not mixing); the doubling map / Bernoulli shift (Bernoulli, hence at the top — K, mixing of all orders, mixing, weakly mixing, ergodic, all at once). Each strictly higher rung adds genuine randomness.
Ergodic < weakly mixing < mixing < K < Bernoulli — a strict tower, with a separating example at every step.
Each implication is one-directional and strict. Beware the common error of treating ergodic, mixing and Bernoulli as synonyms for "random": they are genuinely different, with entropy distinguishing the top rungs and the spectrum distinguishing the bottom ones.