Multifractal Nonrigidity of Topological Markov Chains

Given a multifractal spectrum, we consider the problem of whether it is possible to recover the potential that originates the spectrum. The affirmative solution of this problem would correspond to a “multifractal” classification of dynamical systems, i.e., a classification solely based on the information given by multifractal spectra. For the entropy spectrum on topological Markov chains we show that it is possible to have both multifractal rigidity and multifractal “nonrigidity”, by appropriately varying the Markov chain and the potential defining the spectrum. The “nonrigidity” even occurs in some generic sense. This strongly contrasts to the usual opinion among some experts that it should be possible to recover the potential up to some equivalence relation, at least in some generic sense. Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BD/10154/2002.     

Multifractal Analysis of Local Entropies for Expansive Homeomorphisms with Specification

In the present paper we study the multifractal spectrum of local entropies. We obtain results, similar to those of the multifractal analysis of pointwise dimensions, but under much weaker assumptions on the dynamical systems. We assume our dynamical system to be defined by an expansive homeomorphism with the specification property. We establish the variational relation between the multifractal spectrum and other thermodynamical characteristics of the dynamical system, including the spectrum of correlation entropies. Received: 22 September 1998 / Accepted: 11 December 1998     

Multifractal properties of return time statistics

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions. Comparison with the usual multifractal analysis of measures is presented, and the difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented.     

Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems

 In this paper, we establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation for certain nonhyperbolic systems. We apply our results to Manneville-Pomeau type maps and a piecewise conformal two-dimensional countable Markov map with indifferent periodic points which is related to a complex continued fraction. Received: 6 September 2001 / Accepted: 21 May 2002 Published online: 12 August 2002     

Multifractal Analysis of Asymptotically Additive Sequences

For saturated maps, we effect a complete multifractal analysis of the dimension spectra obtained from asymptotically additive sequences of continuous functions. This includes, for example, the class of maps with the specification property. We consider also the more general cases of ratios of sequences and of multidimensional spectra in which a single sequence is replaced by a vector of sequences. In addition, we establish a conditional variational principle for the topological pressure of a continuous function on the level sets of an asymptotically additive sequence (again in the former general setting). Finally, we apply our results to the dimension spectra of an average conformal repeller. In particular, we obtain almost automatically a conditional variational principle for the Hausdorff dimension of the level sets obtained from an asymptotically additive sequence.     

Multifractal dimensions in QCD cascades

Particle production in small rapidity or angular intervals have fractal structures similar to a Cantor dust. In this paper we present analytical result for multifractal dimensions valid for high energies. For high moments the dimension is given by \(\sqrt {6\alpha _s /\pi } \) . The scaling properties seen in the partonic state are not so well reflected in the hadronic multiplicity moments or factorial moments. We show how to define new observables on the final hadronic state, which do scale well. This means that the multifractal dimensions can be well determined and compared with results from QCD.     

Multifractal dimensions for branched growth

A recently proposed theory for diffusion-limited aggregation (DLA), which models this system as a random branched growth process, is reviewed. Like DLA, this process is stochastic, and ensemble averaging is needed in order to define multifractal dimensions. In an earlier work by Halsey and Leibig, annealed average dimensions were computed for this model. In this paper, we compute the quenched average dimensions, which are expected to apply to typical members of the ensemble. We develop a perturbative expansion for the average of the logarithm of the multifractal partition function; the leading and subleading divergent terms in this expansion are then resummed to all orders. The result is that in the limit where the number of particlesn, the quenched and annealed dimensions areidentical; however, the attainment of this limit requires enormous values ofn. At smaller, more realistic values ofn, the apparent quenched dimensions differ from the annealed dimensions. We interpret these results to mean that while multifractality as an ensemble property of random branched growth (and hence of DLA) is quite robust, it subtly fails for typical members of the ensemble.     

Multifractal clustering in compressible flows

A quantitative relationship is found between the multifractal properties of the asymptotic mass distribution in a random dissipative system and the long-time fluctuations of the local stretching rates of the dynamics. It captures analytically the fine aspects of the strongly intermittent clustering of dynamical trajectories. Applied to a simple compressible hydrodynamical model with known stretching-rate statistics, the relation produces a nontrivial spectrum of multifractal dimensions that is confirmed numerically.     

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.     

Multifractal Analysis of Human Heartbeat in Sleep

We study the dynamical properties of heart rate variability （HRV） in sleep by analysing the scaling behaviour with the multifractal detrended fluctuation analysis method. It is well known that heart rate is regulated by the interaction of two branches of the autonomic nervous system： the parasympathetic and sympathetic nervous systems. By investigating the multifractal properties of light, deep, rapid-eye-movement （REM） sleep and wake stages, we firstly find an increasing multifractal behaviour during REM sleep which may be caused by augmented sympathetic activities relative to non-REM sleep. In addition, the investigation of long-range correlations of HRV in sleep with second order detrended fluctuation analysis presents irregular phenomena. These findings may be helpful to understand the underlying regulating mechanism of heart rate by autonomic nervous system during wake-sleep transitions.     

Multifractal spectrum analysis of nonlinear dynamical mechanisms in China’s agricultural futures markets

Based on Partition Function and Multifractal Spectrum Analysis, we investigated the nonlinear dynamical mechanisms in China’s agricultural futures markets, namely, Dalian Commodity Exchange (DCE for short) and Zhengzhou Commodity Exchange (ZCE for short), where nearly all agricultural futures contracts are traded in the two markets. Firstly, we found nontrivial multifractal spectra, which are the empirical evidence of the existence of multifractal features, in 4 representative futures markets in China, that is, Hard Winter wheat (HW for short) and Strong Gluten wheat (SG for short) futures markets from ZCE and Soy Meal (SM for short) futures and Soy Bean No.1 (SB for short) futures markets from DCE. Secondly, by shuffling the original time series, we destroyed the underlying nonlinear temporal correlation; thus, we identified that long-range correlation mechanism constitutes major contributions in the formation in the multifractals of the markets. Thirdly, by tracking the evolution of left- and right-half spectra, we found that there exist critical points, between which there are different behaviors, in the left-half spectra for large price fluctuations; but for the right-hand spectra for small price fluctuations, the width of those increases slowly as the delay t increases in the long run. Finally, the dynamics of large fluctuations is significantly different from that of the small ones, which implies that there exist different underlying mechanisms in the formation of multifractality in the markets. Our main contributions focus on that we not only provided empirical evidence of the existence of multifractal features in China agricultural commodity futures markets; but also we pioneered in investigating the sources of the multifractality in China’s agricultural futures markets in current literature; furthermore, we investigated the nonlinear dynamical mechanisms based on spectrum analysis, which offers us insights into the underlying dynamical mechanisms in China’s agricultural futures markets.     

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.     

On Multifractal Rigidity

We analyze when a multifractal spectrum can be used to recover the potential. This phenomenon is known as multifractal rigidity. We prove that for a certain class of potentials the multifractal spectrum of local entropies uniquely determines their equilibrium states. This leads to a classification which identifies two systems up to a change of variables.     

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.     

Multifractal Structure of Intermittency in a Developed Ionospheric Turbulence

We present the results of studying the multifractal structure of intermittency in a developed ionospheric turbulence during special experiments on radio-raying of the midlatitude ionosphere by signals from orbital satellites in 2005–2006. It is shown, in particular, that the determination of multidimensional structural functions of the energy fluctuations of received signals permits one to obtain the necessary information on multifractal spectra of the studied process of radio-wave scattering in the ionosphere. Experimental data on multifractal spectra of slow fluctuations in the received-signal energy under conditions of a developed small-scale turbulence are compared with the existing concept of the radio-wave scattering within the framework of the statistical theory of radio-wave propagation in the ionosphere. It is inferred that under conditions of a developed ionospheric turbulence, the multifractal structure of the intermittency of slow fluctuations in the received-signal energy is a consequence of the intermittency of small-scale fluctuations in the electron number density of the ionospheric plasma on relatively large spatial scales of about several ten kilometers. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 6, pp. 485–493, June 2008.     

高能多粒子末态的多重分形维数与动力学起伏强度

研究了多重分形维数与动力学起伏强度的关系,给出了一个能直接描述高能多粒子末态动力学起伏强度的特征参量,并指出了这一参量的适用范围以及在实际高能实验中的应用.     

Multifractal analysis of polyalanines time series

Multifractal properties of the energy time series of short α-helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique (multifractal detrended fluctuation analysis). Estimates for the generalized Hurst exponent h(q) and its associated multifractal exponents τ(q) are obtained for several series generated by numerical simulations of molecular dynamics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects of the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process.     

Universal deterministic statistical independence

For a very special class of symbol sequences, both simple and complicated deterministic chaotic systems universally generate the same multiplicative multifractal generating functions. This new universality is based upon statistical independence that is deterministic in origin and accounts for the evidence that turbulence can mimic certain simple pseudo-random number generators. Our result follows from the construction of classes of symbol sequences that are a generalization of normal numbers and which occur universally for a certain class of dynamical systems. By constructing an example of the required class of symbol sequence, we are able to provide a fully deterministic explanation for thef() spectra of the sort obtained for one-dimensional cuts of turbulence. The application to turbulence in three dimensions is also discussed, as are deterministic noise and the application to experimental time series. The universality limits the extent within which one can infer the details of the dynamics from observedf() spectra.     

Sequential Gibbs Measures and Factor Maps

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of Kempton and Pollicott (Factors of Gibbs measures for full shifts, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011) and Chazottes and Ugalde (On the preservation of Gibbsianness under symbol amalgamation, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011), we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.     

Multifractal method for the instantaneous evaluation of the stream function in geophysical flows

Multifractal or multiaffine analysis is a promising new branch of methods in nonlinear physics for the study of turbulent flows and turbulentlike systems. In this Letter we present a new method based on the multifractal singularity extraction technique, the maximum singular stream-function method (MSSM), which provides a first order approximation to the stream function from experimental data in 2D turbulent systems. The essence of MSSM relies in relating statistical properties associated with the energy cascade in flows with geometrical properties. MSSM is a valuable tool to process sparse collections of data and to obtain instant estimates of the velocity field. We show an application of MSSM to oceanography as a way to obtain the current field from sea surface temperature satellite images; we validate the result with independent dynamical information obtained from sea level measurements.