Orthospectrum and simple orthospectrum rigidity: finiteness and genericity

Abstract

We study the orthospectrum and the simple orthospectrum of compact hyperbolic surfaces with geodesic boundary. We show that there are finitely many hyperbolic surfaces sharing the same simple orthospectrum and finitely many hyperbolic surfaces sharing the same orthospectrum. Then, we show that generic surfaces are determined by their orthospectrum and by their simple orthospectrum. We conclude with the example of the one-holed torus which is determined by its simple orthospectrum. In this second version of the paper an error in the proof of Theorem 3.3 was corrected as well as its consequences on the rest of the paper. The main results remain true.

43 pages, 14 figures