Convergence of the Eckmann and Ruelle algorithm for the estimation of Liapunov exponents

Abstract

We analyze the convergence conditions of the Eckmann and Ruelle algorithm (E.R.A. for the sequel) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence which are related to those ensuring the convergence to the tangent map of the best linear L^{p}-fittings of the action of a mapping f on small balls. Under such conditions, we show how to use E.R.A. to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of E.R.A. for the computation of Liapunov exponents in smooth manifolds which allows us to avoid the problem of detecting the spurious exponents. We prove, for a Borel measurable dynamics f, the existence of Liapunov exponents for the function Sr(x), mapping each point x to the matrix of the best linear Lp-fitting of the action of f on the closed ball of radius r centered at x, and we show how to use E.R.A. to get reliable estimates of the Liapunov exponents of Sr. We also propose a test for checking the differentiability of an empirically observed dynamics.