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We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces.This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of C-valued cohomology into its real and imaginary parts.We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form.We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical.We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstädt for principal strata.We describe all the leaf closures for the horospherical foliation.
Smillie et al. (Fri,) studied this question.