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. 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.

References in corpus (6)

• Ruelle-Perron-Frobenius spectrum for Anosov maps
• Spectral Galerkin methods for transfer operators in uniformly expanding dynamics
• Dynamic mode decomposition for analytic maps
• Variance continuity for Lorenz flows
• Stability and approximation of statistical limit laws for multidimensional piecewise expanding maps
• Higher order asymptotics for large deviations -- Part I

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• A general framework for the rigorous computation of invariant densities and the coarse-fine strategy
• Approximating linear response by nonintrusive shadowing algorithms
• Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems
• Differentiating densities on smooth manifolds
• Lagrange approximation of transfer operators associated with holomorphic data
• Stable laws for random dynamical systems