Generalized dimensions, large deviations and the distribution of rare events

Abstract

Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a chaotic dynamical system, generalized dimensions take on a dynamical meaning, as they provide the rate function for the large deviations of the first hitting time, which is the (average) time required to connect any two different regions in phase space. We prove this result rigorously under a set of stringent assumptions. As a consequence, the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions. Numerical examples, presented along with the theory, suggest that the validity of this technique reaches far beyond the range covered by the theorem.We state our result within the framework of extreme value theory. This approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure. This suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension. We show an application of this technique to experimental data from climate dynamics.

On the Computation of the Extremal Index for Time Series

“…In order to make rigorous computations, we consider a Markov map of the interval T for which the density h is piecewise constant (see for The extremal index is given by the integral in Eq. (35) which has already been computed in [8] and gives θ ≈ 0.5926. The quantitiesα l can also be computed explicitly.…”

“…The interpretation of the EI given in Eq. (8) has been used in applications to time series climate data and it was also emphasized the local character of such an indicator and its strong correlation with the fractal local dimension of the invariant measure [17,7,19].…”

“…In the attempt to describe the synchronisation of coupled map lattices, Faranda et al [15] introduced the neighborhood of the diagonal in the n-dimensional hypercube and defined accordingly the extremal index related to the first time the maps on the lattices become close together. Faranda et al [15] then used that approach to define a new type of observable, first in [20] and successively generalized in [8], which brought to a different interpretation of the extremal index, in terms of Lyapunov exponents instead of periodic behaviors. Let us consider the k−fold (k > 1) direct product (X, µ, T ) k with the direct product map T k = T • · · · • T acting on the product space X k and the product measure µ k = µ • · · · • µ.…”

Extreme value distributions of observation recurrences

“…The operatorP n now decomposes as the sum of a projection along the one dimensional eigenspace associated to the eigenvalue χ n and an operator with a spectral radius exponentially decreasing to zero and which can be neglected in the limit of large n. 6 Remembering this, writingP n h n = χ n h n , with h n converging to h in the L 1 (Leb) norm and replacing into the right hand side of Eq. (16) and after a few manipulations we get, by neglecting higher order terms:…”

Sampling Hyperspheres via Extreme Value Theory: Implications for Measuring Attractor Dimensions

“…Comparing values of d estimated using Euclidean and Mahalanobis distances will allow not only to assess the level of non-randomness in a covariance-stationary systems, but also the relative contribution of linear and nonlinear dependence to the dimensionality reduction. Finally, extreme value theory is a useful tool to derive a certain number of useful quantities in dynamical systems theory: the correlation dimension and the Lyapunov exponents , the generalized dimensions Caby et al (2018) and the synchronization of coupled lattice map . It will then be interesting to evaluate the curse of dimensionality effects on those quantities.…”