Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows

TL;DR

This paper demonstrates rapid mixing and superpolynomial equidistribution for certain hyperbolic flow extensions on tori, under Diophantine conditions, with applications to three-dimensional frame flows.

Contribution

It establishes rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows based on Diophantine properties, advancing understanding of flow dynamics.

Findings

Flow enjoys rapid mixing under Diophantine conditions

Superpolynomial error term in equidistribution of holonomy

Applications to three-dimensional frame flows

Abstract

In this paper, we study mixing rates for $\mathbb{T}^{d}$-extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in $\mathbb{R}^{d+1}$. We prove that the extension flow enjoys rapid mixing, if the associated point is an inhomogeneously Diophantine number. Under the same assumption, we also obtain the superpolynomial equidistribution, namely, a superpolynomial error term in the equidistribution of the holonomy around closed orbits. Lastly, we apply these results to a class of three-dimensional frame flows.