28 research outputs found
Diophantine approximation by orbits of Markov maps
In 1995, Hill and Velani introduced the shrinking targets theory. Given a
dynamical system , they investigated the Hausdorff dimension of sets
of points whose orbits are close to some fixed point. In this paper, we study
the sets of points well-approximated by orbits , where
is an expanding Markov map with a finite partition supported by . The
dimensions of these sets are described using the multifractal properties of
invariant Gibbs measures.Comment: 24 pages, 3 figures; To appear in ETDS, 201
Renewal of singularity sets of statistically self-similar measures
This paper investigates new properties concerning the multifractal structure
of a class of statistically self-similar measures. These measures include the
well-known 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 H\"older 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
measures. Our results are useful to understand the multifractal nature of
various heterogeneous jump processes
A pure jump Markov process with a random singularity spectrum
We construct a non-decreasing 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.Comment: 20 pages, 4 figure
On multifractality and time subordination for continuous functions
We show that if is "homogeneously multifractal" (in a sense we precisely
define), then is the composition of a monofractal function with a time
subordinator (i.e. is the integral of a positive Borel measure
supported by \zu). When the initial function is given, the monofractality
exponent of the associated function is uniquely determined. We study in
details a classical example of multifractal functions , for which we exhibit
the associated functions and . This provides new insights into the
understanding of multifractal behaviors of functions
A survey on prescription of multifractal behavior
International audienceMultifractal behavior has been identified and mathematically established for large classes of functions, stochastic processes and measures. Multifractality has also been observed on many data coming from Geophysics, turbulence, Physics, Biology, to name a few. Developing mathematical models whose scaling and multifractal properties fit those measured on data is thus an important issue. This raises several still unsolved theoretical questions about the prescription of multifractality (i.e. how to build mathematical models with a singularity spectrum known in advance), typical behavior in function spaces, and existence of solutions to PDEs or SPDEs with possible multifractal behavior. In this survey, we gather some of the latest results in this area. Dedicated to Alain Arnéodo, pioneer in the development of wavelet tools for data analysis
A localized Jarnik-Besicovitch Theorem
Fundamental questions in Diophantine approximation are related to the
Hausdorff dimension of sets of the form , where and is the Diophantine
approximation rate of an irrational number . We go beyond the classical
results by computing the Hausdorff dimension of the sets , where is a continuous function. Our theorem applies to
the study of the approximation rates by various approximation families. It also
applies to functions which are continuous outside a set of prescribed
Hausdorff dimension.Comment: 31 pages, Figures are available on our web site
Heterogeneous ubiquitous systems in and Hausdorff dimension
Let be a sequence of , a sequence of positive real numbers converging to 0, and . Let be a positive Borel measure on , and . Consider the limsup-set We show that, under suitable assumptions on the measure , the Hausdorff dimension of the sets can be computed. When , a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of . Our results apply to several classes of multifractal measures . The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) -adic expansion properties, by averages of some Birkhoff sums and by asymptotic behavior of random covering numbers
A survey on prescription of multifractal behavior
International audienceMultifractal behavior has been identified and mathematically established for large classes of functions, stochastic processes and measures. Multifractality has also been observed on many data coming from Geophysics, turbulence, Physics, Biology, to name a few. Developing mathematical models whose scaling and multifractal properties fit those measured on data is thus an important issue. This raises several still unsolved theoretical questions about the prescription of multifractality (i.e. how to build mathematical models with a singularity spectrum known in advance), typical behavior in function spaces, and existence of solutions to PDEs or SPDEs with possible multifractal behavior. In this survey, we gather some of the latest results in this area. Dedicated to Alain Arnéodo, pioneer in the development of wavelet tools for data analysis
p-exponents and p-multifractal spectrum of some lacunary Fourier series
Non UBCUnreviewedAuthor affiliation: Université Paris Est CréteilFacult