Poincaré disk model — exponential divergence, mixing, ergodicity
Poincaré disk (hyperbolic plane)
Geodesic separation log|δ(t)|
Angle distribution (ergodicity)
Negative curvature and chaos: On a surface of constant negative curvature K<0 (hyperbolic plane), geodesics diverge exponentially — adjacent geodesics separate as e√|K| · t. This is the geometric origin of deterministic chaos. The geodesic flow on compact hyperbolic surfaces is Anosov (uniformly hyperbolic), hence ergodic, mixing, and has positive Lyapunov exponent λ=√|K|. The Poincaré disk model visualizes ℍ²: geodesics are arcs of circles orthogonal to the boundary disk.