Geodesic Flow on Negative Curvature
Anosov system: exponential divergence of nearby geodesics, ergodic mixing on hyperbolic surface
Poincaré Upper Half-Plane (H²)
Geodesic Divergence (Anosov property)
Parameters
Curvature K:
-1.00
Geodesics:
8
Initial spread δ:
0.05
Flow time T:
4.00
Surface type:
Poincaré Upper Half-plane
Poincaré Disk
Genus-2 Surface
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Lyapunov λ ≈ √|K|:
—
Mixing time:
—
|K|:
—
Entropy (KS):
—
Trajectories:
0
Hadamard-Perron theorem: geodesic flow on compact surface K<0 is Anosov — stable/unstable foliations split tangent space. Entropy h = √|K| (Liouville 1840s, rigorized by Anosov 1960s).