Geodesics in the hyperbolic plane are circular arcs meeting the boundary perpendicularly; the geodesic flow is chaotic and ergodic.
The Poincaré disk model maps hyperbolic space ℍ² into the unit disk. All geodesics are arcs of circles orthogonal to the boundary ∂𝔻. The geodesic flow on the unit-tangent bundle is mixing and has positive Lyapunov exponent (Hadamard 1898).