Arnold's Cat Map & Ergodic Mixing

Chaotic area-preserving toral automorphism — λ = ln φ (golden ratio)

Cat Map Iterations — Mixing & Recurrence

iteration: 0

After 3 iterations: nearly unrecognizable. After enough steps (~period): returns to cat shape!

Phase Space Mixing

Theory

(x,y) → (x+y, x+2y) mod 1
Matrix: [[1,1],[1,2]] ∈ SL(2,Z)
Eigenvalues: λ± = φ± = (3±√5)/2
Lyapunov exponent: Λ = ln φ ≈ 0.481
Period N: (x,y) returns after ~3N iter (N=40 for 320×320)

Arnold's cat map (1960s): named after the cat image Vladimir Arnold used to illustrate chaos. The matrix has det=1 → area-preserving (symplectic). Integer matrix entries → periodic on rational lattice points.


Anosov diffeomorphism: the cat map is the prototypical example. It has hyperbolic fixed point at origin with stable/unstable manifolds aligned with eigenvectors of the matrix. Ergodic + mixing + exact.


Poincaré recurrence: on a finite N×N grid, all configurations are periodic. A 320×320 grid has period dividing lcm of pixel periods. Most pixels return after 120–240 iterations.