Geodesic flow on negatively curved surfaces is the paradigm of chaos — trajectories diverge exponentially (positive Lyapunov exponent). Arnold's cat map is a discrete Anosov diffeomorphism on the 2-torus with exact hyperbolic structure.
8
2
100
Lyapunov λ₁
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Iteration
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Max separation
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Cat map: (x,y) → (2x+y, x+y) mod 1 | Eigenvalues: λ± = (3±√5)/2 | Lyapunov: ln(φ²) ≈ 0.962
Geodesic flow on H²: ṡ = Jsτ where s = (x,y,θ) ∈ T¹H² | Anosov: stable + unstable foliations transverse everywhere.