The dimension spectrum of some dynamical systems

Collet, Pierre; Lebowitz, Joel; Porzio, Anna

doi:10.1007/bf01206149

Cited by 222 publications

(101 citation statements)

References 17 publications

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“…The existence of the limit (3.12) for binomial cascades has been established in [18] as well as in [38]. It follows also from the following simple observation, which promises wider applicability:…”

Section: Remark 311 (Quenched and Annealed Averages)mentioning

confidence: 77%

Multifractal Processes

Riedi¹

1999

167

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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and self-similar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and self-similar processes. Statistical properties of estimators as well as modelling issues are addressed.AMS Subject classification: Primary 28A80; secondary 37F40. Keywords and phrases: Multifractal analysis, self-similar processes, fractional Brownian motion, Lévy flights, α-stable motion, wavelets, long-range dependence, multifractal subordination. Report Documentation Page Form Approved OMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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Section: Remark 311 (Quenched and Annealed Averages)mentioning

confidence: 77%

Multifractal Processes

Riedi¹

1999

167

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“…the topological pressure P (φ) in (1·5) is equal to 0 (in other words, we replace φ by φ − P (φ)). All the following results are standard results, and can be found in the references we cited in the precedent section: [12,30,8,32,4,26,27].…”

Section: Hitting Timesmentioning

confidence: 99%

“…By an extensive literature (Collet, Lebowitz and Porzio [12], Rand [30], Brown, Michon and Peyrière [8], Simpelaere [32], Barreira, Pesin and Schmeling [4], Pesin and Weiss [26,27]), the multifractal analysis of µ φ can be achieved, i.e. the multifractal spectrum of µ φ can be computed (see Section 2.3 for more details).…”

Section: Introductionmentioning

confidence: 99%

Diophantine approximation by orbits of expanding Markov maps

Liao

Seuret

2012

Ergod. Th. Dynam. Sys.

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Abstract. Given a dynamical system ([0, 1], T ), the distribution properties of the orbits of real numbers x ∈ [0, 1] under T constitute a longstanding problem. In 1995, Hill and Velani introduced the "shrinking targets" theory, which aims at investigating precisely the Hausdorff dimensions of sets whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits {T n x} n≥0 , where T is an expanding Markov map with finite partitions supported by the whole interval [0,1]. The values of the dimensions of sets of well-approximable points are described using the multifractal properties of Gibbs measures invariant under the action of T . This study can be viewed as a moving shrinking targets problem.

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“…Notable examples include the invariant probability distribution on a strange attractor [36,38,43], the distribution of voltage drops across a random resistor network [2,3,44], the distribution of growth probabilities on the boundary of a di usion-limited aggregate [3,4,45] and the spatial distribution of dissipative regions in a turbulent ow [41,[46][47][48]. This formalism lies upon the determination of the so-called f( ) singularity spectrum [36] which characterizes the relative contribution of each singularity of the measure: let S be the subset of points x where the measure of an -box B x ( ), centered at x, scales like (B x ( )) ∼ in the limit → 0 + , then by deÿnition, f( ) is the Hausdor dimension of S : f( ) = dim H (S ).…”

Section: Statistical Analysis Of the Regularity Of Fractal Functions:mentioning

confidence: 99%

“…The multifractal formalism [34][35][36][37][38][39][40][41][42][43] has been originally established to account for the statistical scaling properties of singular measures arising in various situations in physics, chemistry, geology or biology. Notable examples include the invariant probability distribution on a strange attractor [36,38,43], the distribution of voltage drops across a random resistor network [2,3,44], the distribution of growth probabilities on the boundary of a di usion-limited aggregate [3,4,45] and the spatial distribution of dissipative regions in a turbulent ow [41,[46][47][48].…”

Section: Statistical Analysis Of the Regularity Of Fractal Functions:mentioning

confidence: 99%

Thermodynamics of fractal signals based on wavelet analysis: application to fully developed turbulence data and DNA sequences

Arnéodo

Audit

Bacry

et al. 1998

Physica A: Statistical Mechanics and its Applications

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We use the continuous wavelet transform to generalize the multifractal formalism to fractal functions. We report the results of recent applications of the so-called wavelet transform modulus maxima (WTMM) method to fully developed turbulence data and DNA sequences. We conclude by brie y describing some works currently under progress, which are likely to be the guidelines for future research. c 1998 Elsevier Science B.V. All rights reserved From global to local characterization of the regularity of fractal functionsIn many situations in physics as well as in some applied sciences, one is faced to the problem of characterizing very irregular functions [1][2][3][4][5][6][7][8][9][10][11]. The examples range from plots of various kinds of random walks, e.g. Brownian signals [12,13], to ÿnancial timeseries [14][15][16], to geological shapes [1,9,17], to medical time-series [18], to interfaces developing in far from equilibrium growth processes [4,6,11], to turbulent velocity signals [10,19,20] and to "DNA walks" coding nucleotide sequences [21,22]. These functions can be qualiÿed as fractal functions [1,13,[23][24][25] whenever their graphs are fractal sets in R 2 (for our purpose here we will only consider functions from R to R). They are commonly called self-a ne functions since their graphs are similar to themselves when transformed by anisotropic dilations, i.e., when shrinking along the x-axis by a factor followed by a rescaling of the increments of the function by a di erent factor −H

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The dimension spectrum of some dynamical systems

Cited by 222 publications

References 17 publications

Multifractal Processes

Multifractal Processes

Diophantine approximation by orbits of expanding Markov maps

Thermodynamics of fractal signals based on wavelet analysis: application to fully developed turbulence data and DNA sequences

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