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 ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence that 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 ERA to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA 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 $S_r(x)$, mapping each point $x$ to the matrix of the best linear $L^p$-fitting of the action of $f$ on the closed ball of radius $r$ centred at $x$, and we show how to use ERA to get reliable estimates of the Liapunov exponents of $S_r$. We also propose a test for checking the differentiability of an empirically observed dynamics.