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Arnold cat map

Each iteration of the cat map stretches the image along the unstable eigendirection and folds it back by mod 1. On the discrete 64×64 torus the iteration is periodic; on the continuous torus the Lyapunov exponent is ln((3+√5)/2) ≈ 0.962.

Matrix M = [[a, b], [c, d]]

a = 1

b = 1

c = 1

d = 2

Classical Arnold: (a, b, c, d) = (1, 1, 1, 2). det M = 1. Tune for other hyperbolic toral automorphisms.

Grid size N: 64 (Poincaré period k = 48)

Iterations k: 3

Rule

(x_{n+1}, y_{n+1}) = ((1 x_n + 1 y_n) mod N, (1 x_n + 2 y_n) mod N)
matrix M = [[1, 1], [1, 2]]
det M = 1 (area-preserving on the torus ✓)
trace = 3, eigenvalues real and hyperbolic iff |trace| > 2.
Continuous-map Lyapunov: λ = 0.9624
Period on N×N integer torus: 48

Try (a, b, c, d) = (1, 2, 2, 5): still det = 1, larger Lyapunov ⇒ faster mixing. Or (1, 0, 0, 1): identity ⇒ image never changes (det = 1 but λ = 0).

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