Geodesic
Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e151

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In a previous paper, the authors extended Mirzakhani’s (almost-everywhere defined) measurable conjugacy between the earthquake and horocycle flows to a measurable bijection. In this one, we analyze the continuity properties of this map and its inverse, proving that both are continuous at many points and in many directions. This lets us transfer measure convergence between the two systems, allowing us to pull back results from Teichmüller dynamics to deduce analogous statements for the earthquake flow. 
Calderon, Aaron; Farre, James. Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e151. doi: 10.1017/fms.2025.10093
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