Birman and Series showed that the union of all simple closed geodesics
on a finite-area hyperbolic surface is a set of Hausdorff dimension
one; in particular this *Birman-Series set* is nowhere dense and
has Lebesgue measure zero [1].

The **left image** above shows an approximation of a Birman-Series set
consisting of the 88 shortest geodesics on a hyperbolic punctured
torus. Because a long simple geodesic can be approximated
in *C*^{1} by arcs from closed geodesics of moderate
length, this finite union of closed geodesics is a reasonable
approximation of the full set.

The underlying Riemann surface in this example is the one obtained by identifying opposite sides of the unit square and removing the point corresponding to the four corners. A conformal mapping from a fundamental domain for the corresponding Fuchsian group is used to map the geodesics to the unit square.

The **right image** above shows the same collection of homology classes of
closed geodesics on the square Euclidean torus. In contrast to the
sparseness of the hyperbolic case, here the simple closed geodesics
are dense in the unit tangent bundle. While a homology class does not
uniquely determine a closed geodesic in the Euclidean case, in this
image each representatives is chosen to pass through the same set of
Weierstrass points as the corresponding hyperbolic geodesic.

- Hyperbolic geodesics: PDF - High-res PNG
- Euclidean geodesics: PDF - High-res PNG

These images were originally created for a poster session at the workshop "Aspects of hyperbolicity in geometry, topology, and dynamics" at the University of Warwick (celebrating Caroline Series' 60th birthday), July 25-27, 2011. The versions appearing in the final poster use slightly different color, line width, and transparency settings.

- J. S. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed.
*Topology*, 24(2):217-225, 1985.