Kac famously asked if one could hear the shape of a drum or more precisely does the spectrum of the laplacian determine a surface up to isometry. Huber and Selberg showed how this question could be settled using the length spectrum of the surface - that is the lengths of all the closed geodesics.
We consider a similar problem for the ortho spectrum of hyperbolic surfaces with totally geodesic boundary.
The ortho spectrum appeared first in the work of Meyerhoff and Basmajian and is the set of all arcs perpendicular to the boundary. We will first discuss the geometric invariants that are known to be determined by the orthospectrum
- the perimeter
- the area
- the entropy of the geodesic flow.
Thereafter we present a result with H. Masai that in general the ortho spectrum does not
determine the systolic length but that there are only finitely many possibilities for the the systolic length for a given ortho spectrum.
In fact we show that, up to isometry,
there are only finitely many
hyperbolic structures on a surface
that share a given ortho spectrum.
This extends results of McKean and Wolpert for closed geodesics.