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Peter G. Doyle and Juan Pablo Rossetti
Isospectral hyperbolic surfaces have matching geodesics
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| Published: |
June 5, 2008 |
| Keywords: |
Isospectrality, closed geodesic, holonomy, almost conjugate, hyperbolic surface, Selberg trace formula, flat orbifold, prime geodesic theorem |
| Subject: |
Primary: 58J53; Secondary: 11F72, 20F67, 53C22 |
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Abstract
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.
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| Acknowledgements
Partially supported by DFG Sonderforschungsbereich 647, Humboldt University, Berlin.
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| Author information
Peter G. Doyle:
Dartmouth College
doyle@math.dartmouth.edu
Juan Pablo Rossetti:
FaMAF-CIEM, Univ. Nac. Córdoba
rossetti@mate.uncor.edu
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