Chaotic stretching & folding: the mechanism of fluid mixing
Phase space (unit square): colored blobs under iteration
3
0.40
2
2000
Baker map:
(x,y) → (x/λ, λy) if x<λ
(x,y) → ((x−λ)/(1−λ), λ+(1−λ)y) if x≥λ
Lyapunov exp: λ₁ = —
Topol. entropy: h = —
Area preserved: ✓
KS entropy = —
Baker map: a prototype of chaotic mixing. Each iteration stretches in x by 1/λ (λ<½) and compresses in y, then cuts and stacks — like kneading dough. After n iterations, the initial blob is stretched by (1/λ)^n and has 2^n tendrils, achieving exponential mixing. Lyapunov exponent λ₁ = −ln λ. The invariant measure is uniform (area-preserving) — Liouville's theorem for Hamiltonian chaos.