Set of Birman Series

The Birman-Series theorem , named after Joan Birman and Caroline Series , is a theorem of hyperbolic geometry . It says that, unlike in the Euclidean geometry of the torus, the geodesics are not dense on a hyperbolic surface.

Set of Birman Series

Let it be a hyperbolic surface of finite area, i.e. a 2-dimensional manifold with a Riemannian metric of constant negative curvature and finite area. ${\displaystyle S}$

For was the amount of surveyors from a maximum of transversal self-intersections and the amount of points on a geodesic in lying. ${\displaystyle k\geq 0}$ ${\displaystyle G_{k}}$ ${\displaystyle S}$ ${\displaystyle k}$ ${\displaystyle S_{k}\subset S}$ ${\displaystyle G_{k}}$

Then there is a nowhere dense subset of and it has Hausdorff dimension 1. ${\displaystyle S_{k}}$ ${\displaystyle S}$

Simple closed curves

The special case means in particular that the simple closed geodesics are nowhere close together. Even this simplest case distinguishes hyperbolic surfaces significantly from Euclidean surfaces: on a Euclidean torus, the simple closed geodesics are dense everywhere. ${\displaystyle k=0}$

literature

• Joan Birman , Caroline Series : Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology 24 (1985) no. 2, 217-225. pdf
• Albert Fathi : Expansiveness, hyperbolicity and Hausdorff dimension. Comm. Math. Phys. 126 (1989) no. 2, 249-262.
• Jenya Sapir: Non-simple geodesics on surfaces , Stanford University 2014
• Anna Lenzhen, Juan Souto: Variations on a theorem of Birman and Series , pdf

Web links

• Simple Geodesics and the Birman-Series set (images)