Decay of correlations for certain isometric extensions of Anosov flows

Abstract

We establish exponential decay of correlations of all orders for locally G-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable distributions of the base Anosov flow are $C^1$ . This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by infinitesimal transitivity groups used by Dolgopyat, from which we obtain contraction properties for a class of ‘twisted’ symbolic transfer operators.

Towards Brin’s conjecture on frame flow ergodicity: new progress and perspectives

“…• Exponential mixing for the frame flow: Using representation theory, exponential mixing was proved by Moore [25] on hyperbolic manifolds, and reproved recently in dimension n = 3 by Guillarmou-Küster [18] using semiclassical analysis. Rapid mixing of isometric extensions of diffeomorphisms was studied by Dolgopyat [17] and more recently, by Siddiqi [33] in the case of flows. However, due to low regularity issues of the stable/unstable foliation, going beyond constant curvature is still an open question.…”

Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

“…He proved the equivalence between an infinitesimal non-integrability condition and the exponential mixing of compact group extensions of expanding maps on closed manifolds. In [Sid22], Siddiqi considered the compact extensions of a certain class of Anosov flows, where he translated the accessibility properties of the extension into Dolgopyat's non-integrability condition. Besides dynamical approaches, there are works using further analytic tools to study frame flows.…”