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

We analyse the convergence conditions of the Eckmann and Ruelle algorithm ( ERA) used to estimate the Liapunov exponents, for the tangent map, of an er godic measure, invariant under a smooth dynamical system. We find sufficien t conditions for this convergence that are related to those ensuring the co nvergence 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 ER A to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA for the computation of Liapuno v exponents in smooth manifolds, which allows us to avoid the problem of de tecting the spurious exponents. We prove, for a Borel measurable dynamics f, the existence of Liapunov expo nents for the function S-r(x), mapping each point x to the matrix of the be st Linear L-p-fitting of the action of f on the closed bah of radius r cent red at x, and we show how to use ERA to get reliable estimates of the Liapu nov exponents of S-r. We also propose a test for checking the differentiabi lity of an empirically observed dynamics.