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Inside singularity sets of random Gibbs measures
 J. Stat. Phys
, 2005
"... Abstract. 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 ..."
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Cited by 13 (9 self)
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Abstract. 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 selfsimilar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes. 1.
A PURE JUMP MARKOV PROCESS WITH A RANDOM SINGULARITY SPECTRUM
"... Abstract. We construct a nondecreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on ..."
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Cited by 11 (5 self)
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Abstract. We construct a nondecreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems. 1.
Ubiquity and large intersections properties under digit frequencies constraints
, 2009
"... digit frequencies constraints ..."
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Renewal of singularity sets of statistically selfsimilar measures
, 2006
"... Abstract. This paper investigates new properties concerning the multifractal structure of a class of statistically selfsimilar measures. These measures include the wellknown Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifr ..."
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Cited by 7 (4 self)
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Abstract. This paper investigates new properties concerning the multifractal structure of a class of statistically selfsimilar measures. These measures include the wellknown Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these 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 selfsimilar measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes. 1.
DIOPHANTINE APPROXIMATION BY ORBITS OF EXPANDING MARKOV MAPS
"... 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 ..."
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Cited by 4 (1 self)
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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 wellapproximated 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 wellapproximable 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. 1.
RENEWAL OF SINGULARITY SETS OF RANDOM SELFSIMILAR MEASURES
 APPLIED PROBABILITY TRUST (9 FEBRUARY 2007)
, 2007
"... This paper investigates new properties concerning the multifractal structure of a class of random selfsimilar measures. These measures include the wellknown Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure o ..."
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Cited by 4 (2 self)
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This paper investigates new properties concerning the multifractal structure of a class of random selfsimilar measures. These measures include the wellknown Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these 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 selfsimilar measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.
THE SINGULARITY SPECTRUM OF THE INVERSE OF COOKIECUTTERS
"... Abstract. Gibbs measures µ on cookiecutter sets are the archetype of multifractal measures on Cantor sets. In this article we compute the singularity spectrum of the inverse measure of µ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism, and ..."
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Cited by 3 (0 self)
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Abstract. Gibbs measures µ on cookiecutter sets are the archetype of multifractal measures on Cantor sets. In this article we compute the singularity spectrum of the inverse measure of µ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism, and its L qspectrum possesses one point of non differentiability. The results rely on heterogeneous ubiquity theorems. 1.
A LOCALIZED JARNIKBESICOVITCH THEOREM
"... Abstract. Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x ∈ R: δx = δ}, where δ ≥ 1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of ..."
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Cited by 1 (1 self)
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Abstract. Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x ∈ R: δx = δ}, where δ ≥ 1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x ∈ R: δx = f(x)}, where f is a continuous function. Our theorem applies to the study of the approximation exponents by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension. 1.
ProjectTeam SISYPHE SIgnals and SYstems in PHysiology and Engineering
"... c t i v it y e p o r t ..."
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"... This paper investigates new properties concerning the multifractal structure of random selfsimilar measures. The class of measures to which our results apply includes the wellknown Mandelbrot multiplicative cascades [39], sometimes called independent random cascades. The case of another important c ..."
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This paper investigates new properties concerning the multifractal structure of random selfsimilar measures. The class of measures to which our results apply includes the wellknown Mandelbrot multiplicative cascades [39], sometimes called independent random cascades. The case of another important class, the random Gibbs measures, is treated in