Hyperbolic Geometry
The Poincaré disk is a model of the hyperbolic plane H²: the open unit disk with metric ds² = 4(dx²+dy²)/(1-r²)² giving constant curvature K=−1.
Geodesics are circular arcs meeting the boundary at right angles (or diameters). The {p,q} tessellation tiles H² with p-gons meeting q at each vertex — infinitely many fit inside the disk.
Geodesic flow on negatively curved surfaces is Anosov: uniformly hyperbolic, ergodic, mixing, and Bernoulli — the prototype of mathematical chaos. Nearby trajectories diverge exponentially at rate equal to the curvature.
hyperbolic geometry
Anosov flow
ergodic theory
Möbius transform