Horocycle flow - Knowlpedia

Definition

On an oriented hyperbolic surface $M=\Gamma\backslash\mathbb H^2$ , identify the unit tangent bundle with $\Gamma\backslash PSL(2,\mathbb R)$ . A horocycle flow is right translation by one of the unipotent subgroups

n_s^+=\begin{pmatrix}1&s\\0&1\end{pmatrix}, \qquad n_s^-=\begin{pmatrix}1&0\\s&1\end{pmatrix}.

The two choices give the unstable and stable horocycle flows.

Geometric picture

Projected orbits trace horocycles: curves orthogonal to all geodesics converging to the same point on the ideal boundary. The geodesic flow expands one horocycle direction and contracts the other.

Compact quotients

For a compact hyperbolic surface, each horocycle flow is uniquely ergodic. This recurrence forces the endpoint set of geodesics avoiding a fixed nonempty open set to have holes at all relevant scales.

References

Gustav A. Hedlund, “Fuchsian groups and transitive horocycles,” Duke Mathematical Journal 2 (1936), 530–542. DOI record.