796 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
Quantitative recurrence properties in conformal iterated function systems
Let be a countable index set and be a
conformal iterated function system on satisfying the open set
condition. Denote by the attractor of . With each sequence
is associated a unique point . Let denote the set of points of with unique coding, and
define the mapping by . In this paper, we consider the quantitative recurrence
properties related to the dynamical system . More precisely, let
be a positive function and
where is the th Birkhoff sum associated with the potential .
In other words, contains the points whose orbits return close to
infinitely often, with a rate varying along time. Under some conditions, we
prove that the Hausdorff dimension of is given by , where is the pressure function and is the
derivative of . We present some applications of the main theorem to
Diophantine approximation.Comment: 25 page
Multifractal properties of typical convex functions
We study the singularity (multifractal) spectrum of continuous convex
functions defined on . Let be the set of points at which
has a pointwise exponent equal to . We first obtain general upper bounds
for the Hausdorff dimension of these sets , for all convex functions
and all . We prove that for typical/generic (in the sense of
Baire) continuous convex functions , one has for all and in addition, we obtain that the set is empty if . Also, when is typical,
the boundary of belongs to
Measures and functions with prescribed homogeneous multifractal spectrum
In this paper we construct measures supported in with prescribed
multifractal spectrum. Moreover, these measures are homogeneously multifractal
(HM, for short), in the sense that their restriction on any subinterval of
has the same multifractal spectrum as the whole measure. The spectra
that we are able to prescribe are suprema of a countable set of step
functions supported by subintervals of and satisfy for all
. We also find a surprising constraint on the multifractal spectrum
of a HM measure: the support of its spectrum within must be an
interval. This result is a sort of Darboux theorem for multifractal spectra of
measures. This result is optimal, since we construct a HM measure with spectrum
supported by . Using wavelet theory, we also build HM functions
with prescribed multifractal spectrum.Comment: 34 pages, 6 figure
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