Lp(μ) estimation of tangent maps

Abstract

We analyze under what conditions the best Lp(μ)-linear fittings of the action of a mapping f on small balls give reliable estimates of the tangent map Df. We show that there is an inverse relationship between the conditions on the regularity, in terms of local densities, of the measure μ and the smoothness of the mapping f which are required to ensure the goodness of the estimates. The above results can be applied to the estimation of tangent maps in two empirical settings: from finite samples of a given probability distribution on IRⁿ and from finite orbits of smooth dynamical systems. As an application of the results of this paper we obtain sufficient conditions on the measure μ to ensure the convergence of Eckmann and Ruelle algorithm for computing the Liapunov exponents of smooth dynamical systems.