Geodesic Flow on the Hyperbolic Plane (K = −1)

Poincaré disk model:
ds² = 4(dx²+dy²)/(1−r²)²

Geodesics = circular arcs orthogonal to ∂D.

K = −1 → exponential divergence:
d(t) ~ e^t · d(0) (Anosov flows)

Entropy h = 1 (top. entropy)
Mixing: correlations decay exponentially.

Geodesic flow is uniformly hyperbolic → Anosov → ergodic (Hopf 1939)

Geodesic spread: —