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In a previous paper, the authors extended Mirzakhani's (almost-everywheredefined) measurable conjugacy between the earthquake and horocycle flows to ameasurable bijection. In this one, we analyze the continuity properties of thismap and its inverse, proving that both are continuous at many points and inmany directions. This lets us transfer measure convergence between the twosystems, allowing us to pull back results from Teichm{\"u}ller dynamics todeduce analogous statements for the earthquake flow.

Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow