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.

Spectral analysis of climate dynamics with operator-theoretic approaches

“…In particular, numerical representations of P are often not unitary. Nevertheless, numerical schemes such as projected restrictions of P or U onto subspaces of H spanned by locally supported or globally supported basis functions have been highly successful 17 , 23 , 29 , 59 and in certain settings, convergence results for the spectrum and eigenfunctions have been proven 14 , 15 , 22 , 60 – 62 . In these schemes, the spectrum of the approximate P is contained in the unit disk , rather than lying on the unit circle .…”

“…In these schemes, the spectrum of the approximate P is contained in the unit disk , rather than lying on the unit circle . This addition of noise, which may also be done theoretically, for example by convolution with a stochastic kernel 15 , 62 , 63 , is frequently harnessed to easily select the most important eigenvalues from the typically infinite collection σ e ( P ), namely those eigenvalues with large magnitude (close to 1).…”

Spectral analysis of climate dynamics with operator-theoretic approaches

“…In particular, numerical representations of P are often not unitary. Nevertheless, numerical schemes such as projected restrictions of P or U onto subspaces of H spanned by locally supported or globally supported basis functions have been highly successful 17,23,59,60 and in certain settings, convergence results for the spectrum and eigenfunctions have been proven 14,15,22,61,62 . In these schemes, the spectrum of the approximate P is contained in the unit disk {z ∈ C : |z| ≤ 1}, rather than lying on the unit circle {z ∈ C : |z| = 1}.…”

“…In these schemes, the spectrum of the approximate P is contained in the unit disk {z ∈ C : |z| ≤ 1}, rather than lying on the unit circle {z ∈ C : |z| = 1}. This addition of noise, which may also be done theoretically, for example by convolution with a stochastic kernel 15,62,63 , is frequently harnessed to easily select the most important eigenvalues from the typically infinite collection σ e (P), namely those eigenvalues with large magnitude (close to 1).…”

Statistical Stability for Deterministic and Random Dynamical Systems

“…This theory is then applied to smooth random expanding maps on the circle, and the stability of some basic statistical properties is deduced with respect to fibre-wise deterministic perturbations and a Fourier-analytic numerical method. Some of this research has been published in [1][2][3].…”