We prove an extension of Basmajian's identity to $n$-Hitchin representationsof compact bordered surfaces. For $n=3$, we show that this identity has ageometric interpretation for convex real projective structures analogous toBasmajian's original result. As part of our proof, we demonstrate that, withrespect to the Lebesgue measure on the Frenet curve associated to a Hitchinrepresentation, the limit set of an incompressible subsurface of a closedsurface has measure zero. This generalizes a classical result in hyperbolicgeometry. Finally, we recall the Labourie-McShane extension of theMcShane-Mirzakhani identity to Hitchin representations and note a closeconnection to Basmajian's identity in both the hyperbolic and the Hitchinsettings.
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机译：我们证明了Basmajian身份的扩展到紧边框表面的$ n $ -Hitchin表示。对于$ n = 3 $，我们证明该身份具有类似于Basmajian原始结果的凸实投影结构的年龄度量解释。作为证明的一部分，我们证明，相对于与Hitchin表示法相关的Frenet曲线上的Lebesgue测度，封闭曲面的不可压缩次表面的极限集的测度为零。这概括了双曲线几何中的经典结果。最后，我们回想起了McShane-Mirzakhani身份对Hitchin表示形式的Labourie-McShane扩展，并注意到在双曲和Hitchinsettings中，Basmajian的身份紧密相关。
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