Fourier approximation of the statistical properties of Anosov maps on tori

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 Gouëzel and Liverani (2006 Ergod. Theor. Dyn. Syst. 26 189–217). By extending our previous work in Crimmins and Froyland (2019 Ann. Henri Poincaré 20 3113–3161), 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 methods 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. We consequently obtain the first rigorous numerical methods for estimating the variance and rate function for Anosov maps.

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