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Extreme value theory for singular measures.

Valerio Lucarini1, Davide Faranda, Giorgio Turchetti

  • 1Klimacampus, Institute of Meteorology, University of Hamburg, Grindelberg 5, 20144 Hamburg, Germany. valerio.lucarini@zmaw.de

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Summary

Extreme value analysis of dynamical systems reveals that extremes follow a generalized extreme value distribution, linked to the attractor's information dimension. This method uncovers geometric properties of dynamical systems.

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Area of Science:

  • Dynamical Systems Theory
  • Statistical Mechanics
  • Chaos Theory

Background:

  • Dynamical systems often exhibit complex behavior governed by invariant measures.
  • Understanding the extreme values of observables is crucial for characterizing system dynamics.
  • Singular measures present unique challenges in analyzing system properties.

Purpose of the Study:

  • To analytically and numerically investigate extreme values of observables in dynamical systems with invariant singular measures.
  • To establish the statistical distribution of these extreme values and its relation to system geometry.
  • To explore the utility of extreme value analysis in characterizing the geometric structure of attractors.

Main Methods:

  • Analytical study of observables based on the distance from a point on the attractor.
  • Application of the block maxima approach for extreme value analysis.
  • Numerical simulations on low-dimensional maps (Cantor set, Sierpinski triangle, Lozi, Hénon maps).

Main Results:

  • Extreme values are distributed according to the generalized extreme value distribution.
  • Parameters of this distribution are functions of the attractor's information dimension.
  • Numerical results for iterated function systems show excellent agreement with theoretical predictions.
  • Strange attractors exhibit slower convergence but yield statistically consistent estimates.

Conclusions:

  • Extreme value analysis effectively captures fundamental information about the geometric structure of dynamical system attractors.
  • The chosen observables act as a 'magnifying glass' to probe attractor geometry.
  • The generalized extreme value distribution provides a robust framework for analyzing extremes in these systems.