Asymptotic expansion of correlation functions for $\mathbb{Z}^d$ covers of hyperbolic flows

TL;DR

This paper proves detailed asymptotic expansions for correlation functions in certain hyperbolic flow extensions, including models like the periodic Lorentz gas and geodesic flows on abelian covers, advancing understanding of their statistical properties.

Contribution

It establishes asymptotic expansion of correlation functions for $bZ^d$ extensions of hyperbolic flows, covering new examples such as the $bZ^2$ periodic Lorentz gas.

Findings

Correlation functions admit expansions of all orders.

Results apply to models like Lorentz gas and geodesic flows on abelian covers.

Enhances understanding of statistical behavior in hyperbolic dynamical systems.

Abstract

We establish expansion of every order for the correlation function of sufficiently regular observables of $\mathbb Z^d$ extensions of some hyperbolic flows. Our examples include the $\mathbb Z^2$ periodic Lorentz gas and geodesic flows on abelian covers of compact manifolds with negative curvature.