Isospectral hyperbolic surfaces have matching geodesics

PETER S. DOYLE; JUAN PABLO ROSSETTI

New York Journal of Mathemtatics

Año: 2008 vol. 14 p. 193 - 204

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics.Restricted to orientable surfaces, this result reduces to Huber’s theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.