THE THERMODYNAMICS OF FRACTALS REVISITED WITH WAVELETS

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.

[1]  U. Frisch,et al.  Wavelet transforms of self-similar processes , 1991 .

[2]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[3]  E. Bacry,et al.  Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  L. Sander,et al.  Diffusion-limited aggregation , 1983 .

[5]  E. Bacry,et al.  Singularity spectrum of fractal signals from wavelet analysis: Exact results , 1993 .

[6]  R. Benzi,et al.  Wavelet analysis of a Gaussian Kolmogorov signal , 1993 .

[7]  Mitchell J. Feigenbaum.  Some characterizations of strange sets , 1987 .

[8]  M. Nelkin.  What do we know about self-similarity in fluid turbulence? , 1989 .

[9]  Y. Couder,et al.  Direct observation of the intermittency of intense vorticity filaments in turbulence. , 1991, Physical review letters.

[10]  Schram,et al.  Generalized dimensions from near-neighbor information. , 1988, Physical review. A, General physics.

[11]  B. Derrida,et al.  Universal metric properties of bifurcations of endomorphisms , 1979 .

[12]  Yves Meyer,et al.  Wavelets and Applications , 1992 .

[13]  U. Frisch.  Ou en est la Turbulence Developpée , 1985 .

[14]  Y. Sinai,et al.  Feigenbaum universality and the thermodynamic formalism , 1984 .

[15]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[16]  S. Shenker,et al.  Quasiperiodicity in dissipative systems: A renormalization group analysis , 1983 .

[17]  Pierre Gilles Lemarié,et al.  Les Ondelettes en 1989 , 1990 .

[18]  Shlomo Havlin,et al.  Dynamic mechanisms of disorderly growth: Recent approaches to understanding diffusion limited aggregation , 1990 .

[19]  Jensen,et al.  Scaling structure and thermodynamics of strange sets. , 1987, Physical review. A, General physics.

[20]  New trends in nonlinear dynamics and pattern-forming phenomena : the geometry of nonequilibrium , 1990 .

[21]  Argoul,et al.  Self-similarity of diffusion-limited aggregates and electrodeposition clusters. , 1988, Physical review letters.

[22]  Argoul,et al.  Golden mean arithmetic in the fractal branching of diffusion-limited aggregates. , 1992, Physical review letters.

[23]  Argoul,et al.  Wavelet analysis of the self-similarity of diffusion-limited aggregates and electrodeposition clusters. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[24]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[25]  Paul Manneville,et al.  Dissipative Structures and Weak Turbulence , 1995 .

[26]  M. Goodchild.  Fractals and the accuracy of geographical measures , 1980 .

[27]  Eric D. Siggia,et al.  Numerical study of small-scale intermittency in three-dimensional turbulence , 1981, Journal of Fluid Mechanics.

[28]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[29]  G. Broggi,et al.  Measurement of the dimension spectrum ƒ(α): Fixed-mass approach , 1988 .

[30]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[31]  Bessis,et al.  Mellin transforms of correlation integrals and generalized dimension of strange sets. , 1987, Physical review. A, General physics.

[32]  James P. Sethna,et al.  Universal properties of the transition from quasi-periodicity to chaos , 1983 .

[33]  Itamar Procaccia,et al.  Phase transitions in the thermodynamic formalism of multifractals. , 1987 .

[34]  Romain Murenzi,et al.  Wavelet Transform of Fractal Aggregates , 1989 .

[35]  U. Frisch,et al.  Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade , 1989, Nature.

[36]  P. Dimotakis,et al.  The lift of a cylinder executing rotary motions in a uniform flow , 1993, Journal of Fluid Mechanics.

[37]  P. Grassberger.  Generalized dimensions of strange attractors , 1983 .

[38]  R. Jensen,et al.  Direct determination of the f(α) singularity spectrum , 1989 .

[39]  Jensen,et al.  Time ordering and the thermodynamics of strange sets: Theory and experimental tests. , 1986, Physical review letters.

[40]  Steven A. Orszag,et al.  Turbulence: Challenges for Theory and Experiment , 1990 .

[41]  Freeman J. Dyson,et al.  An Ising ferromagnet with discontinuous long-range order , 1971 .

[42]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[43]  S. Shenker.  Scaling behavior in a map of a circle onto itself: Empirical results , 1982 .

[44]  Françoise Argoul,et al.  Structural five-fold symmetry in the fractal morphology of diffusion-limited aggregates , 1992 .

[45]  T. Bohr,et al.  Fractal «aggregates» in the complex plane , 1988 .

[46]  E. Bacry,et al.  BEYOND CLASSICAL MULTIFRACTAL ANALYSIS USING WAVELETS: UNCOVERING A MULTIPLICATIVE PROCESS HIDDEN IN THE GEOMETRICAL COMPLEXITY OF DIFFUSION LIMITED AGGREGATES , 1993 .

[47]  A. Arneodo,et al.  Fractal dimensions and (a) spectrum of the Hnon attractor , 1987 .

[48]  Stéphane Mallat,et al.  Singularity detection and processing with wavelets , 1992, IEEE Trans. Inf. Theory.

[49]  Mitchell J. Feigenbaum,et al.  The transition to aperiodic behavior in turbulent systems , 1980 .

[50]  H. Stanley,et al.  On growth and form : fractal and non-fractal patterns in physics , 1986 .

[51]  Benoit B. Mandelbrot,et al.  Multifractal measures, especially for the geophysicist , 1989 .

[52]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[53]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[54]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[55]  Jensen,et al.  Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets. , 1987, Physical review. A, General physics.

[56]  A. Grossmann,et al.  Cycle-octave and related transforms in seismic signal analysis , 1984 .

[57]  Giorgio Mantica,et al.  Inverse problems in fractal construction: moment method solution , 1990 .

[58]  Pierre Collet,et al.  The dimension spectrum of some dynamical systems , 1987 .

[59]  P. Dimotakis,et al.  Stochastic geometric properties of scalar interfaces in turbulent jets , 1991 .

[60]  P. Cvitanović.  Universality in Chaos , 1989 .

[61]  L. Sander,et al.  Diffusion-limited aggregation, a kinetic critical phenomenon , 1981 .

[62]  Uncovering a multiplicative process in one-dimensional cuts of diffusion-limited aggregates , 1995 .

[63]  Michał Misiurewicz,et al.  Dimension of invariant measures for maps with exponent zero , 1985, Ergodic Theory and Dynamical Systems.

[64]  Wentian Li,et al.  ABSENCE OF 1/f SPECTRA IN DOW JONES DAILY AVERAGE , 1991 .

[65]  Howard M. Taylor,et al.  On the Distribution of Stock Price Differences , 1967, Oper. Res..

[66]  C. Peng,et al.  Long-range correlations in nucleotide sequences , 1992, Nature.

[67]  Steven A. Orszag,et al.  Intermittent vortex structures in homogeneous isotropic turbulence , 1990, Nature.

[68]  Y. Sinai.  GIBBS MEASURES IN ERGODIC THEORY , 1972 .

[69]  From global, ag la Kolmogorov 1941, scaling to local multifractal scaling in fully developed turbulence , 1993 .

[70]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[71]  Vicsek,et al.  Multifractality of self-affine fractals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[72]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier , 1974, Journal of Fluid Mechanics.

[73]  T. Vicsek,et al.  Multifractal Geometry of Diffusion-Limited Aggregates , 1990 .

[74]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .

[75]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[76]  E. Bacry,et al.  The Multifractal Formalism Revisited with Wavelets , 1994 .

[77]  Comment on "Self-similarity of diffusion-limited aggregates and electrodeposition clusters" , 1989, Physical review letters.

[78]  M. Feigenbaum.  The universal metric properties of nonlinear transformations , 1979 .

[79]  Yoshiki Kuramoto,et al.  Chaos and Statistical Methods , 1984 .

[80]  G. Dangelmayr,et al.  The Physics of Structure Formation , 1987 .

[81]  F. Anselmet,et al.  High-order velocity structure functions in turbulent shear flows , 1984, Journal of Fluid Mechanics.

[82]  L. Pietronero,et al.  Fractals' physical origin and properties , 1989 .

[83]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[84]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[85]  Jensen,et al.  Extraction of underlying multiplicative processes from multifractals via the thermodynamic formalism. , 1989, Physical review. A, General physics.

[86]  A. Arneodo,et al.  Wavelet transform of multifractals. , 1988, Physical review letters.

[87]  Javier Jiménez,et al.  The structure of intense vorticity in isotropic turbulence , 1993, Journal of Fluid Mechanics.

[88]  Marcel Lesieur,et al.  Turbulence and Coherent Structures , 1991 .

[89]  P. Grassberger,et al.  Dimensions and entropies of strange attractors from a fluctuating dynamics approach , 1984 .

[90]  A. Vincent,et al.  The spatial structure and statistical properties of homogeneous turbulence , 1991, Journal of Fluid Mechanics.

[91]  Antonio Politi,et al.  Intrinsic oscillations in measuring the fractal dimension , 1984 .

[92]  C. Meneveau,et al.  The multifractal nature of turbulent energy dissipation , 1991, Journal of Fluid Mechanics.

[93]  Benoit B. Mandelbrot,et al.  Fractals in physics: Squig clusters, diffusions, fractal measures, and the unicity of fractal dimensionality , 1984 .

[94]  S. Fauve,et al.  Pressure fluctuations in swirling turbulent flows , 1993 .

[95]  Françoise Argoul,et al.  Uncovering Fibonacci sequences in the fractal morphology of diffusion-limited aggregates , 1992 .

[96]  Charles Meneveau,et al.  Measurement of ƒ(α) from scaling of histograms, and applications to dynamical systems and fully developed turbulence , 1989 .

[97]  Paul Meakin,et al.  Growth Patterns in Physical Sciences and Biology , 1993 .

[98]  E. Tirapegui.  Field Theory, Quantization and Statistical Physics , 1981 .

[99]  M. Brachet.  Direct simulation of three-dimensional turbulence in the Taylor–Green vortex , 1991 .

[100]  E. Bacry,et al.  Solving the Inverse Fractal Problem from Wavelet Analysis , 1994 .

[101]  Michael Ghil,et al.  Turbulence and predictability in geophysical fluid dynamics and climate dynamics , 1985 .

[102]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[103]  J. Sethna,et al.  Universal Transition from Quasiperiodicity to Chaos in Dissipative Systems , 1982 .

[104]  Uriel Frisch,et al.  A simple dynamical model of intermittent fully developed turbulence , 1978, Journal of Fluid Mechanics.

[105]  Roberto Benzi,et al.  On the multifractal nature of fully developed turbulence and chaotic systems , 1984 .

[106]  B. Huberman,et al.  Fluctuations and simple chaotic dynamics , 1982 .

[107]  Prasad,et al.  Multifractal nature of the dissipation field of passive scalars in fully turbulent flows. , 1988, Physical review letters.

[108]  M. Holschneider.  On the wavelet transformation of fractal objects , 1988 .

[109]  Jensen,et al.  Direct determination of the f( alpha ) singularity spectrum and its application to fully developed turbulence. , 1989, Physical review. A, General physics.

[110]  Bambi Hu,et al.  Introduction to real-space renormalization-group methods in critical and chaotic phenomena , 1982 .

[111]  Leonard A. Smith,et al.  Lacunarity and intermittency in fluid turbulence , 1986 .

[112]  Mitchell J. Feigenbaum,et al.  Scaling spectra and return times of dynamical systems , 1987 .

[113]  P. Grassberger,et al.  Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors , 1988 .

[114]  Victor Martin-Mayor,et al.  Field Theory, the Renormalization Group and Critical Phenomena , 1984 .

[115]  Y. Gagne,et al.  Velocity probability density functions of high Reynolds number turbulence , 1990 .

[116]  B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

[117]  E. Bacry,et al.  Wavelets and multifractal formalism for singular signals: Application to turbulence data. , 1991, Physical review letters.

[118]  Luciano Pietronero,et al.  FRACTALS IN PHYSICS , 1990 .

[119]  T. Vicsek,et al.  Dynamics of fractal surfaces , 1991 .

[120]  Yves Meyer,et al.  Progress in wavelet analysis and applications , 1993 .

[121]  T. Vicsek.  Fractal Growth Phenomena , 1989 .

[122]  Stéphane Mallat,et al.  Characterization of Self-Similar Multifractals with Wavelet Maxima , 1994 .