Stability and Approximation of Statistical Limit Laws for Multidimensional Piecewise Expanding Maps

Abstract

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani [30] we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-Hölder spaces.

Lagrange approximation of transfer operators associated with holomorphic data

“…There exists by now a considerable body of literature concerned with the application of Ulam-type methods to approximate the leading eigenvalue and eigenfunction as well as subleading eigenvalues of transfer operators, including those arising from higher-dimensional expanding or hyperbolic maps, see [F1,BlK,DJ,BlKL,F2,BahB,GN,CrF1] to name but a few. For dynamical system with higher regularity, the speed of convergence of projection-based methods can be improved by choosing projections onto subspaces spanned by functions of higher smoothness, see [Liv] for a discussion of a general strategy or [BalH] for an approach using wavelets.…”

Fourier approximation of the statistical properties of Anosov maps on tori

“…[13]) to relate the spectral data of an analytically twisted Perron-Frobenius operator to statistical properties of the system. The stability these properties then follows from the stability of the spectrum of the twisted Perron-Frobenius operator [8].…”

“…Rigorous numerical schemes for estimating the variance have so far been considered for C k expanding circle maps [33], piecewise analytic expanding interval maps [18], Lasota-Yorke maps [2,8], the intermittent Liverani-Saussol-Vaienti interval map [2], and piecewise expanding multidimensional maps [8]. All of these methods produce computable bounds for the approximation error, except [8] which demonstrates convergence of the numerical approximation with increasing numerical resolution.…”

“…Rigorous numerical schemes for estimating the variance have so far been considered for C k expanding circle maps [33], piecewise analytic expanding interval maps [18], Lasota-Yorke maps [2,8], the intermittent Liverani-Saussol-Vaienti interval map [2], and piecewise expanding multidimensional maps [8]. All of these methods produce computable bounds for the approximation error, except [8] which demonstrates convergence of the numerical approximation with increasing numerical resolution. We will show that our new Fourier-based constructions and the anisotropic spaces of [14] fit into the general stability framework developed in [8], which yields stability of the CLT variance for a broad class of deterministic, stochastic, and numerical perturbations.…”

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].…”