# Fourier approximation of the statistical properties of Anosov maps on tori

## TL;DR

This paper develops Fourier-analytic methods to rigorously approximate the statistical properties of Anosov maps on tori, including variance and rate functions, under various perturbations, especially numerical schemes.

## Contribution

It introduces new Fourier-based techniques for stable approximation of statistical properties of Anosov maps, including the first rigorous scheme for peripheral spectral data without mollification.

## Key findings

- Established stability of statistical properties under general perturbations.
- Developed two Fourier-analytic approximation schemes for spectral data.
- Provided the first rigorous estimates of variance and rate function for Anosov maps.

## Abstract

We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gou\"ezel and Liverani. By extending our previous work in [8], we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourier-analytic method for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the Perron-Frobenius operator of an Anosov map without mollification. Using the two schemes we obtain the first rigorous estimates of the variance and rate function for Anosov maps.