Extreme value theory for singular measures.

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.

[1]  E. Zelenhasić,et al.  A Stochastic Model for Flood Analysis , 1970 .

[2]  Catherine Nicolis,et al.  Extreme value distributions in chaotic dynamics , 1995 .

[3]  Valerio Lucarini,et al.  Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems , 2011, 1110.0176.

[4]  Andrei Török,et al.  Extreme value theory for non-uniformly expanding dynamical systems , 2012 .

[5]  Holger Kantz,et al.  Reactions to extreme events: Moving threshold model , 2006 .

[6]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[7]  Claudia Klüppelberg,et al.  Extreme Value Theory in Finance , 2008 .

[8]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[9]  P. Friederichs,et al.  Statistical Downscaling of Extreme Precipitation Events Using Censored Quantile Regression , 2007 .

[10]  R. Katz,et al.  Extreme events in a changing climate: Variability is more important than averages , 1992 .

[11]  M. Barnsley,et al.  Iterated function systems and the global construction of fractals , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Marc B. Parlange,et al.  STATISTICS OF EXTREMES: MODELING ECOLOGICAL DISTURBANCES , 2005 .

[13]  F. Ledrappier,et al.  The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula , 1985 .

[14]  H. Lilliefors.  On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown , 1967 .

[15]  R. Katz.  Extreme value theory for precipitation: sensitivity analysis for climate change , 1999 .

[16]  Jürgen Kurths,et al.  Recurrence plots for the analysis of complex systems , 2009 .

[17]  Jose D. Salas,et al.  Regional Frequency Analysis of Extreme Precipitation in Northeastern Colorado and Fort Collins Flood of 1997 , 2002 .

[18]  G. Nicolis,et al.  Extreme events in deterministic dynamical systems. , 2006, Physical review letters.

[19]  Richard L. Smith.  Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone , 1989 .

[20]  S. Vannitsem,et al.  Statistical properties of the temperature maxima in an intermediate order Quasi-Geostrophic model , 2007 .

[21]  Vladimir Kossobokov,et al.  Extreme events: dynamics, statistics and prediction , 2011 .

[22]  F. Ledrappier,et al.  Some relations between dimension and Lyapounov exponents , 1981 .

[23]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[24]  F. Ledrappier,et al.  The metric entropy of diffeomorphisms Part II: Relations between entropy, exponents and dimension , 1985 .

[25]  E. Gumbel.  The Return Period of Flood Flows , 1941 .

[26]  Donald A. Dawson,et al.  Estimation of dimension for spatially distributed data and related limit theorems , 1989 .

[27]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[28]  Giorgio Turchetti,et al.  Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution , 2011, 1103.0889.

[29]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[30]  Bessis,et al.  Mellin transforms of correlation integrals and generalized dimension of strange sets. , 1987, Physical review. A, General physics.

[31]  C. Ferro,et al.  Robust extremes in chaotic deterministic systems. , 2009, Chaos.

[32]  Ana Cristina Moreira Freitas,et al.  Extreme Value Laws in Dynamical Systems for Non-smooth Observations , 2010, 1006.3276.

[33]  Lennart Carleson,et al.  The Dynamics of the Henon Map , 1991 .

[34]  Badii,et al.  Renyi dimensions from local expansion rates. , 1987, Physical review. A, General physics.

[35]  Anthony C. Davison,et al.  Modelling Excesses over High Thresholds, with an Application , 1984 .

[36]  C. Cornell.  Engineering seismic risk analysis , 1968 .

[37]  D. Sornette,et al.  Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes , 1995, cond-mat/9510035.

[38]  Lai-Sang Young,et al.  Sinai-Bowen-Ruelle measures for certain Hénon maps , 1993 .

[39]  P. Burton.  Seismic risk in southern Europe through to India examined using Gumbel's third distribution of extreme values , 1979 .

[40]  Matthew Nicol,et al.  Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps , 2011, Ergodic Theory and Dynamical Systems.

[41]  P. Ruti,et al.  Accessing extremes of mid-latitudinal wave activity: methodology and application , 2009 .

[42]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[43]  Richard L. Smith.  Threshold Methods for Sample Extremes , 1984 .

[44]  Pierre Collet,et al.  Statistics of closest return for some non-uniformly hyperbolic systems , 1999, Ergodic Theory and Dynamical Systems.

[45]  Jorge Milhazes Freitas,et al.  On the link between dependence and independence in extreme value theory for dynamical systems , 2008 .

[46]  Ana Cristina Moreira Freitas,et al.  Hitting time statistics and extreme value theory , 2008, 0804.2887.

[47]  Giorgio Turchetti,et al.  Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets , 1988 .

[48]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.