Upper half-plane H² — geodesics are semicircles & vertical lines
Poincaré disk model — exponential separation of nearby geodesics
Geodesic Setup
Anosov Properties
Lyapunov exponent1.000
Sep. at T—
Mixing rateexponential
Entropy h1 (constant curv.)
In H² (constant curvature -1), geodesics are semicircles orthogonal to ℝ. Nearby geodesics diverge exponentially: d(t) ~ d(0)·e^t.
This is the Anosov property: geodesic flow on a compact hyperbolic surface is uniformly hyperbolic with metric entropy h = 1 (KAM-free, Bernoulli mixing).
This is the Anosov property: geodesic flow on a compact hyperbolic surface is uniformly hyperbolic with metric entropy h = 1 (KAM-free, Bernoulli mixing).