Inside Singularity Sets of Random Gibbs Measures

We evaluate the scale at which the multifractal structure of some random Gibbs measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.     

Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure

The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. We propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S＆P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose sealing ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.     

Three-dimensional wavelet-based multifractal method: the need for revisiting the multifractal description of turbulence dissipation data

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issued from 3D isotropic turbulence simulations. We comment on the need to revisit previous box-counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures.     

The multifractal analysis of Gibbs measures: Motivation,mathematical foundation,and examples

We first motivate the study of multifractals. We then present a rigorous mathematical foundation for the multifractal analysis of Gibbs measures invariant under dynamical systems. Finally we effect a complete multifractal analysis for several classes of hyperbolic dynamical systems. (c) 1997 American Institute of Physics.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.     

Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional turbulence velocity and vorticity data

We use singular value decomposition techniques to generalize the wavelet transform modulus maxima method to the multifractal analysis of vector-valued random fields. The method is calibrated on synthetic multifractal 2D vector measures and monofractal 3D fractional Brownian vector fields. We report the results of some application to the velocity and vorticity fields issued from 3D isotropic turbulence simulations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.     

Joint multifractal analysis based on wavelet leaders

Mutually interacting components form complex systems and these components usually have long-range cross-correlated outputs. Using wavelet leaders, we propose a method for characterizing the joint multifractal nature of these long-range cross correlations; we call this method joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on dual binomial measures with multifractal cross correlations and bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable of detecting cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and determine intriguing joint multifractal behavior.     

Multifractal Formalism for Almost all Self-Affine Measures

We conduct the multifractal analysis of self-affine measures for “almost all” family of affine maps. Besides partially extending Falconer’s formula of L ^q -spectrum outside the range 1 < q ≤ 2, the multifractal formalism is also partially verified.     

Multifractal Analysis for Lyapunov Exponents on Nonconformal Repellers

For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem nor the classical thermodynamic formalism. We use instead a nonadditive topological pressure to characterize the topological entropy of each level set. This prevents us from estimating the complexity of the level sets using the classical Gibbs measures, which are often one of the main ingredients of multifractal analysis. Instead, we avoid even equilibrium measures, and thus in particular g-measures, by constructing explicitly ergodic measures, although not necessarily invariant, which play the corresponding role in our work.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BPD/12108/2003.     

Log-Infinitely Divisible Multifractal Processes

 We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson ``product of cylindrical pulses' [7]. Their construction involves some ``continuous stochastic multiplication' [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non-degeneracy, convergence of the moments and multifractal scaling. Received: 8 July 2002 / Accepted: 17 December 2002 Published online: 14 April 2003 Communicated by A. Connes     

Discrete wavelet approach to multifractality

A new and simple method is presented to study local scaling properties of measures defined on regular and fractal supports. The method, based on a discrete wavelet analysis (WA), complements the well-known multifractal analysis (MA) extensively used in many physical problems. The present wavelet approach is particularly suitable for problems where the multifractal analysis does not provide conclusive results, as e.g. in the case of measures corresponding to Anderson localized wave-functions in one-dimension. Examples of different types of measures are also discussed which illustrate the usefulness of the WA to classify non-multifractal measures according to additional characteristic exponents, which can not be obtained within the MA.     

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.     

A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.     

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.     

On Measures Driven by Markov Chains

We study measures on \([0,1]\) which are driven by a finite Markov chain and which generalize the famous Bernoulli products.We propose a hands-on approach to determine the structure function \(\tau \) and to prove that the multifractal formalism is satisfied. Formulas for the dimension of the measures and for the Hausdorff dimension of their supports are also provided. Finally, we identify the measures with maximal dimension.     

Large Deviations for Countable to One Markov Systems

In this paper, we study large deviation properties for countable to one Markov systems associated to weak Gibbs measures for non-Hölder potentials. Furthermore, we establish multifractal large deviation laws for countable to one piecewise conformal Markov systems, which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency. We apply our results to higher-dimensional number theoretical transformations.     

Factorization of joint multifractality

Complex systems often exhibit multifractal characteristics in various forms. The study of the joint fluctuation of multifractal objects, referred to as joint multifractality, is presented in this work. We use the joint partition function approach [C. Meneveau, et al., Phys. Rev. A. 41 (1990) 894] to show that joint multifractality admits a factorization into a common factor related to the notion of relative multifractality studied by Riedi and Scheuring [R.H. Riedi, I. Scheuring, Fractals 5 (1997) 153] and a remainder term related to the individual multifractality. We demonstrated our ideas using binomial measures and applied to the fluctuation of financial data.     

Empirical evidences of a common multifractal signature in economic, biological and physical systems

Recent developments in microcanonical multifractal formalism have lead to a sensible improvement in the numerical techniques for the determination of the multifractal characteristics of real signals. With the aid of these techniques, we have found empirical evidence of a common multifractal signature in six very different systems, ranging from stock market time series to sea surface temperature records. These systems are not only found to be multifractal, but their singularity spectra are coincident. We propose an explanation of this striking coincidence in terms of a cascade process and analyze its consequences.