The cat map is a paradigmatic hyperbolic automorphism of the 2-torus T² = ℝ²/ℤ²:
(x', y') = (x + y, x + 2y) mod 1
General form: M = [[1, a], [b, ab+1]] with det(M) = 1.
Eigenvalues λ > 1 and 1/λ — stable and unstable manifolds stretch exponentially.
Lyapunov exponent = log λ > 0. The map is Anosov: every trajectory is hyperbolic.
Despite chaos, the map is periodic on rational initial conditions — for a grid of
N×N points the cat image reconstructs exactly after a finite recurrence period (up to ~3N steps).
Left panel: point cloud evolution. Right panel: the classic "cat" being mixed.