a mixing transformation
Mixing strengthens ergodicity from "asymptotic independence on average" to "asymptotic independence outright". Intuitively, a drop of ink stirred into water is mixing: after enough stirring, the fraction of ink in any region equals its overall fraction, regardless of where the ink started. The dynamics forgets initial conditions in a strong, region-by-region sense.
Formally, (Omega, F, P, T) is (strongly) mixing if for all A, B in F, P(T^(-n) A intersect B) -> P(A) P(B) as n -> infinity. Reading it: the chance of being in B now and in A after n steps factorises into independent chances in the limit, so the far future is asymptotically independent of the present. In the Koopman / spectral picture this is the decay of correlations <U^n f, g> - E[f] E[g] -> 0 for all f, g in L^2 — the dynamical system has no almost-periodic structure left at infinity. Compare ergodicity, which only asks the Cesaro average of P(T^(-n) A intersect B) to tend to P(A) P(B); mixing demands the sequence itself converge, which is strictly stronger.
Mixing matters because it is the qualitative signature of genuine chaos and the gateway to limit theorems for dynamical systems: under mixing (with quantitative rates of correlation decay) one can prove central limit theorems and invariance principles for Birkhoff sums, where independence is replaced by fast enough forgetting. Honest cautions: mixing is strictly stronger than weak mixing, which is strictly stronger than ergodicity, so all three are distinct. The irrational rotation is ergodic but not mixing — its correlations oscillate and never decay. And the bare definition gives no rate; quantitative mixing (exponential or polynomial decay of correlations) is an extra, often hard, property that the limit theorems actually require.
The doubling map T(x) = 2x mod 1 on [0,1) (equivalently the one-sided Bernoulli shift) is mixing: for intervals A, B the preimage T^(-n) A becomes a union of 2^n tiny copies spread uniformly across [0,1), so P(T^(-n) A intersect B) -> P(A) P(B). Correlations of smooth observables decay exponentially.
The doubling map mixes; the irrational rotation does not — same recurrence, very different forgetting.
Mixing is a limit statement with no built-in speed. The central limit theorems for dynamical systems need a rate (e.g. summable or exponential correlation decay); plain mixing does not deliver that on its own.