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

Abstract

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 Teichm{ü}ller dynamics to deduce analogous statements for the earthquake flow.

Final version, to appear in Forum of Math, Sigma

References in corpus (4)

• Length statistics of random multicurves on closed hyperbolic surfaces
• Counting hyperbolic multi-geodesics with respect to the lengths of individual components
• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• The asymmetry of Thurston's earthquake flow