Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

TL;DR

This paper proves exponential mixing of frame flows for geometrically finite hyperbolic manifolds with cusps, using symbolic coding, Dolgopyat's method, and large deviation techniques to handle cusp-related challenges.

Contribution

It extends exponential mixing results to manifolds with cusps by developing new methods to address cusp-related technical difficulties.

Findings

Exponential decay of correlations for frame flows with cusps.

Validation of Dolgopyat's method in the presence of cusps.

Large deviation properties for symbolic recurrence in hyperbolic manifolds.

Abstract

As a final work to establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, this paper focuses on the case with cusps. To prove this, we utilize the countably infinite symbolic coding of the geodesic flow of Li-Pan and perform a frame flow version of Dolgopyat's method \`{a} la Sarkar-Winter and Tsujii-Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large proper subset of the limit set. To overcome this, we use a large deviation property for symbolic recurrence to the large subset. It is proved by studying the combinatorics of cusp excursions and using an effective renewal theorem as in the work of Li; the latter uses the exponential decay of the transfer operators for the geodesic flow of Li-Pan.