CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

November 3, 2020 from 16:00 to 17:00 (Montreal/EST time) Zoom meeting

Stability and approximation of statistical limit laws

Seminar presented by Gary Froyland (UNSW Sydney, Australia)

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. We prove 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 then apply these results to piecewise expanding maps in one and multiple dimensions. This theory can be extended to uniformly hyperbolic maps and in this setting we develop two new Fourier-analytic methods to provide the first rigorous estimates of the variance and rate function for Anosov maps.  This is joint work with Harry Crimmins.