= Huber spectral equivalence theorem
{c}
= Huber's theorem
{c}
{synonym}
For <closed> orientable <hyperbolic surfaces> of curvature $-1$, the <spectrum> of the <Laplace-Beltrami operator>, including multiplicities, determines the unmarked <length spectrum> of <closed geodesics>, and conversely. Count either oriented <geodesics> throughout or unoriented ones throughout. Primitive lengths with multiplicities determine the lengths of all iterates, and these determine the primitive lengths recursively.<Huber's theorem for hyperbolic orbisurfaces by Emily B. Dryden and Alexander Strohmaier>[Section 1, classical Huber theorem and oriented length-spectrum convention]{ref}
The <Selberg trace formula> connects the two lists: the spectral side sums functions of <eigenvalues>, while the geometric side sums explicit contributions from closed-geodesic lengths. Equality of either complete list forces equality of the other. The theorem concerns these spectral data; distinct surfaces can still have equal spectra.
For nonorientable <hyperbolic surfaces>, the trace contributions distinguish orientation-preserving and orientation-reversing <geodesics>. Their separate counts, rather than an unspecified undifferentiated convention, belong in the extended statement.<Isospectral hyperbolic surfaces have matching geodesics by Peter G. Doyle and Juan Pablo Rossetti>[Theorem 1 and Sections 2–3]{ref}
Back to article page
