Multifractal formalism from large deviations
Journal of Mathematical Analysis and Applications, vol. 555, pp. 130139
Abstract
It has often been observed that the multifractal formalism and the large deviation principles are intimately related. In fact, multifractal formalism was heuristically derived using the ideas of large deviations. In many cases where multifractal properties have been rigorously established, the corresponding large deviation results hold as well. This naturally raises the question of under what conditions can the multifractal formalism be directly deduced from the corresponding large deviation results. In this paper, we take the first steps in establishing a systematic program for deriving multifractal formalism directly from large deviations. We consider a sequence { X n } n ∈ N of measurable functions on a metric space Ω, satisfying a large deviation principle, and consider the level sets K α = { ω : lim n → ∞ X n ( ω ) n = α }. Under additional technical conditions, typical in multifractal analysis, we estimate the entropy spectrum of K α directly from the large deviation rate function associated with the sequence { X n } , and demonstrate that many known results in multifractal formalism can be immediately recovered within this framework. Moreover, we strongly believe that the additional conditions can be weakened further In the final section, we outline several directions to advance the program of deriving multifractal formalism directly from large deviations.
Authors 3
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Affiliation as printed
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, the Netherlands
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Leiden University · University of Amsterdam
Affiliation as printed
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, the Netherlands
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, the Netherlands
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Affiliation as printed
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China