Exponential Mixing of Frame Flows for Geometrically Finite Hyperbolic Manifolds

by Prof. Jialun Li (CNRS & École polytechnique)

Monday Mar 6, 2023, 4:00 PM → 5:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Let M be a geometrically finite hyperbolic manifold, that is, a hyperbolic manifold with a fundamental domain consisting of a finitely-sided polyhedron. There exists a unique measure on the unit tangent bundle invariant under the geodesic flow with maximal entropy, and we consider its lift to the frame bundle. In joint work with Pratyush Sarkar and Wenyu Pan, we prove that the frame flow is exponentially mixing with respect to this measure. To establish exponential mixing, we base ourselves on the countable coding of the flow and a version of Dolgopyat's method, à la Sarkar-Winter and Tsujii-Zhang. To overcome the difficulty of the fractal structure in applying Dolgopyat's method, we prove a large deviation property for symbolic recurrence to the large subsets.