Parameters
Key Theorem
Hopf (1939): Geodesic flow on a compact surface of negative curvature is ergodic — time averages equal space averages for a.e. trajectory.
Lyapunov exponent λ > 0: neighboring geodesics diverge as e^{λt}.
This is the simplest proven example of deterministic chaos.
Lyapunov exponent λ > 0: neighboring geodesics diverge as e^{λt}.
This is the simplest proven example of deterministic chaos.
Poincaré Disk Model
Hyperbolic plane: metric ds² = 4(dx²+dy²)/(1−r²)²
Geodesics = circular arcs orthogonal to boundary (or diameters).
Ideal boundary = {|z|=1} — points at "infinity" (never reached in finite time).
Geodesics = circular arcs orthogonal to boundary (or diameters).
Ideal boundary = {|z|=1} — points at "infinity" (never reached in finite time).
Divergence
Lyapunov: ~0.00