Tangent measures and Lp estimation of tangent maps

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Facultad de Ciencias Económicas y Empresariales. Decanato
Google Scholar
Research Projects
Organizational Units
Journal Issue
We analyze under what conditions the best Lp- 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 relationslúp between the conditions on the regularity, in terms of local densities, of the underlying measure 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 fiuite samples of a given probability distribution on mn and from fiuite orbits of smooth dynamical systems.
En este artículo se analiza bajo qué condiciones las mejores estimaciones lineales en norma Lp- para la acción de una función f sobre bolas de radio pequeño, proporcionan estimaciones fiables de la aplicación tangente Df. Se comprueba que existe una relación inversa entre las condiciones de regularidad, en términos de densidades locales, de la medida subyacente y la suavidad que se requiere a la transformación f para asegurar la bondad de las estimaciones. Los resultados anteriores pueden aplicarse para estimar la aplicación tangente en dos situaciones que se presentan en el trabajo empírico: a partir de muestras finitas de una distribución de probabilidad en IRn y a partir de órbitas finitas de sistemas dinámicos diferenciables.
Billingsley, P.; Convergence of Prabability Measures, John Wiley (1968). Cutler, C.D.; Connecting Ergodicity and Dimension in Dynamical Systems, Ergodic Th. Dynam. Syst., 10, (1990),451-462. Davis, P.J.; Interpolation and Appraximation. Blaisdell Publishing Company (1963). Eckmann, J.P.; S.O. Kamphorst, D. Ruelle and S. CHiberto, Liapunov Exponents from Time Series, Physical Review; A,34,6,(1989),4971-4979. Federer, H.; Geometric Measure Theory, Springer-Verlag, Berlín-Heidelberg-New York, 1969. Mattila, P.; Geometry 01 Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge University Press (1995). Mera, M.E. and M. Morán, Convergence of the Eckmann and Ruelle Algorithm for the Estimation of Liapunov Exponents, manuscript in preparation. Pesin, Y. and H. Weiss, A Multifractal Analysis of Equilibrium Measures for Conformal Expanding Maps and Markov Moran Geometric Constructions, preprint. Preiss, D.; Geometry of Measures in IRn: Distribution, Rectifiability and Densities, Annals of Mathematics, 125 (1987), 537-643. Rogers, C.A. and S.J. Taylor, Functions Continuous and Singular with respect to a Hausdorff Measure, Mathematika, 8 (1961), 1-31. Rudin, W.; Análisis Real y Complejo, Alhambra (1985). Stein E.M. Singular Integrals and Differentiability Praperties of Functíons, Princeton University Press. Princeton, New Jersey (1970). Tricot, C.; Two definitions ofFractional Dimension, Math. Prac. Camb. Phil. Soc., 91, (1982), 57-74. Young, L.S.; Dimension, Entropy and Lyapunov Exponents, Ergod. Theor.& Dyn. Sys., 2 (1982), 109.