Geodesic Billiards: Hadamard & Hyperbolic Chaos

Negative curvature • exponential divergence • Sinai billiard • Lyapunov exponents

Billiard Table

Geodesic equation:
ẍ + Γᵅᵦᵧ ẋᵝẋᵞ = 0

Negative curvature → chaos
K < 0 ⟹ λ_max > 0

Lyapunov: |δx(t)| ~ e^{λt}
Sinai (1970): hyperbolic
billiards are ergodic
Bounces: 0
Lyapunov est:
Billiards in Sinai's table (square with circular obstacle) are provably chaotic — nearby trajectories diverge exponentially. This is the simplest model of deterministic chaos through geometric focusing, foundational to ergodic theory and quantum chaos.