weak mixing
Weak mixing sits strictly between ergodicity and (strong) mixing. It captures systems that mix "in density" — correlations decay to zero along almost all times even if they refuse to decay at every single time. It is the property that rules out the one obstruction the irrational rotation displays: persistent quasi-periodic oscillation.
There are several equivalent definitions, and their coincidence is the heart of the theory. (i) Density mixing: (1/n) sum_(k=0)^(n-1) | P(T^(-k) A intersect B) - P(A) P(B) | -> 0 for all A, B — correlations vanish in Cesaro average of absolute values, so they are small for a density-one set of times. (ii) Product ergodicity: the product system (Omega x Omega, P x P, T x T) is ergodic. (iii) Spectral / continuous-spectrum criterion: the Koopman operator U has no eigenvalues other than 1 (and that one simple), i.e. the only measurable eigenfunctions are constants; there is no almost-periodic component. The equivalence of (i)-(iii) — particularly that weak mixing equals having continuous spectrum — is a striking structural fact.
Weak mixing matters as the right dividing line: it is exactly the absence of nontrivial "discrete spectrum" (eigenvalues), which is what a rotation has and a chaotic map lacks. It implies ergodicity (take A = B invariant) and is implied by mixing, but is strictly weaker than mixing — there exist weakly mixing systems that are not mixing (and these are in fact generic in a Baire-category sense). And it is strictly stronger than ergodicity, the irrational rotation being ergodic but not weakly mixing (it has lots of eigenvalues e^(2 pi i n alpha)). So weak mixing is the precise statement "ergodic with no rotational, almost-periodic part".
The product test in action: the irrational rotation T fails weak mixing because T x T on the torus is not ergodic — the anti-diagonal {(x, y) : y - x = const} is invariant. A weakly mixing T, by contrast, has T x T ergodic, so its self-correlations cannot conspire to keep an invariant set in the product.
Weak mixing = ergodicity of the self-product = continuous spectrum = no almost-periodic part.
Density-one is the crucial weakening: weak mixing allows correlations to be large at a sparse (density-zero) set of times. Strong mixing forbids that entirely, which is why weak mixing is genuinely weaker.