Generalized dimensions, large deviations and the distribution of rare events

PHYSICA D-NONLINEAR PHENOMENA

卷:400

Article

Caby, Theophile^1,2  Faranda, Davide^3,4,5  Mantica, Giorgio^1,6,7  Vaienti, Sandro²  Yiou, Pascal^3,4 

[1] Univ Insubria, Dipartimento Sci & Alta Tecnol, Ctr Nonlinear & Complex Syst, Como, Italy

[2] Univ Toulon & Var, Aix Marseille Univ, CNRS, CPT, F-13009 Marseille, France

[3] IPSL, UVSQ, CNRS, UMR 8212,CEA,Lab Sci Climat & Environm, F-91191 Gif Sur Yvette, France

[4] Univ Paris Saclay, F-91191 Gif Sur Yvette, France

[5] London Math Lab, 8 Margravine Gardens, London W6 8RH, England

[6] INFN, Sez Milano, Rome, Italy

[7] Indam, Grp Nazl Fis Matemat, Rome, Italy

关键词: Generalized multifractal dimensions;    Hitting times;    Large deviations;    Extreme value theory;    Climate dynamics;   

DOI  :  10.1016/j.physd.2019.06.009

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free