Flat traces for a random partially hyperbolic map

TL;DR

This paper studies the statistical behavior of flat traces of transfer operators for a class of random partially hyperbolic maps, showing convergence to Gaussian distributions up to a certain Ehrenfest time.

Contribution

It introduces a new analysis of flat traces for random partially hyperbolic maps and establishes their convergence to Gaussian laws under specific conditions.

Findings

Flat traces converge to Gaussian distributions in law.

Convergence holds up to an Ehrenfest time depending on regularity.

Results extend understanding of transfer operators in random dynamical systems.

Abstract

We consider a $\mathbb R/\mathbb Z$ extension of an Anosov diffemorphism of a compact Riemannian manifold by a random function $\tau$ and show that the flat traces of the transfer operator, reduced with respect to frequency in the fibers, converge in law towards Gaussians, up to an Ehrenfest time that decreases with the regularity of $\tau$.