Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Abstract

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

“…When δ > d/2, Theorem 1.1 was proved by Mohammadi-Oh [33] and Edwards-Oh [19] using the representation theory of L 2 (M) and the spectral gap of Laplace operator [28]. When is convex cocompact, i.e., geometrically finite without parabolic elements, Theorem 1.1 and its corollaries were proved by Naud [34], Stoyanov [47] and Sarkar-Winter [44] building on the work of Dolgopyat [18]. Therefore, the main contribution of our work lies in the groups with small critical exponent and with parabolic elements, completing the story of exponential mixing of the geodesic flow on a geometrically finite hyperbolic manifold.…”

“…We prove this by using the coding of the geodesic flow constructed in this paper and then performing a frame flow version of Dolgopyat's method. The crucial cancellations of the summands of the transfer operators twisted by holonomy are obtained from the local non-integrability condition and the non-concentration property of Sarkar-Winter [44]. But the challenge in the presence of cusps is that the latter holds only on a certain good subset.…”

Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$

“…In this section, we describe the reduction from Property (2) of Theorem 4.5 to Theorem 10.3 which is the main technical theorem in our setting associated to Dolgopyat's method [Dol98]. These techniques are now well developed and we mainly follow [OW16,Sto11,SW21].…”

Decay of correlations for certain isometric extensions of Anosov flows

“…Theorem A is analogous to a result of Dolgopyat, who showed in [12] that accessible compact group extensions of discrete-time expanding dynamical systems are exponentially mixing, and whose techniques we make heavy use of in our proof. Our approach follows the strategy Winter used in [17] to establish exponential mixing for frame flows over convex cocompact hyperbolic manifolds with Dolgopyat's techniques. We begin by constructing a symbolic model for the extension flow, establishing uniform local non-integrability estimates for this symbolic model and using arguments employed by Dolgopyat in both 1384 S. Siddiqi [11,12] to obtain bounds for the spectrum of certain 'twisted' transfer operators.…”