Ushijima coordinates and rigidity of the (simple) orthospectrum

In 1993, Basmajian introduces the orthospectrum: The multiset of lengths of orthogeodesics on a hyperbolic surface with boundary. We ask ourselves "Does the orthospectrum determine up to isometry a hyperbolic surface ?". In 2023, Masai and McShane gave an anwser and a result of rigidity for surfaces with only 1 boundary component. On another note, hyperbolic surfaces live in the Teichmüller space, which is usually described with Fenchel-Nielsen coordinates. In this talk, we will see how for hyperbolic surfaces with boundary we can use a different set of coordinates to study the rigidity of the orthospectrum and the simple orthospectrum: Ushijima coordinates.