137 research outputs found
Minkowski content and fractal Euler characteristic for conformal graph directed systems
We study the (local) Minkowski content and the (local) fractal Euler
characteristic of limit sets of conformal graph directed
systems (cGDS) . For the local quantities we prove that the logarithmic
Ces\`aro averages always exist and are constant multiples of the
-conformal measure. If is non-lattice, then also the non-average
local quantities exist and coincide with their respective average versions.
When the conformal contractions of are analytic, the local versions
exist if and only if is non-lattice. For the non-local quantities the
above results in particular imply that limit sets of Fuchsian groups of
Schottky type are Minkowski measurable, proving a conjecture of Lapidus from
1993. Further, when the contractions of the cGDS are similarities, we obtain
that the Minkowski content and the fractal Euler characteristic of exist if
and only if is non-lattice, generalising earlier results by Falconer,
Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar
subsets of that satisfy the open set condition.Comment: 34 page
Large deviation asymptotics for continued fraction expansions
We study large deviation asymptotics for processes defined in terms of
continued fraction digits. We use the continued fraction digit sum process to
define a stopping time and derive a joint large deviation asymptotic for the
upper and lower fluctuation process. Also a large deviation asymptotic for
single digits is given.Comment: 15 page
Strong laws of large number for intermediately trimmed Birkhoff sums of observables with infinite mean
We consider dynamical systems on a finite measure space fulfilling a spectral
gap property and Birkhoff sums of a non-negative, non-integrable observable.
For such systems we generalize strong laws of large numbers for intermediately
trimmed sums only known for independent random variables. The results split up
in trimming statements for general distribution functions and for regularly
varying tail distributions. In both cases the trimming rate can be chosen in
the same or almost the same way as in the i.i.d. case. As an example we show
that piecewise expanding interval maps fulfill the necessary conditions for our
limit laws. As a side result we obtain strong laws of large numbers for
truncated Birkhoff sums.Comment: 37 page
A distributional limit law for the continued fraction digit sum
We consider the continued fraction digits as random variables measured with
respect to Lebesgue measure. The logarithmically scaled and normalized
fluctuation process of the digit sums converges strongly distributional to a
random variable uniformly distributed on the unit interval. For this process
normalized linearly we determine a large deviation asymptotic.Comment: 14 pages, 1 figur
Fractal analysis for sets of non-differentiability of Minkowski's question mark function
In this paper we study various fractal geometric aspects of the Minkowski
question mark function We show that the unit interval can be written as
the union of the three sets ,
, and does
not exist and The main result is that the Hausdorff
dimensions of these sets are related in the following way.
Here, refers to the level set of the
Stern-Brocot multifractal decomposition at the topological entropy
of the Farey map and
denotes the Hausdorff dimension of the measure of maximal entropy of the
dynamical system associated with The proofs rely partially on the
multifractal formalism for Stern-Brocot intervals and give non-trivial
applications of this formalism.Comment: 22 pages, 2 figure
Regularity of multifractal spectra of conformal iterated function systems
We investigate multifractal regularity for infinite conformal iterated
function systems (cIFS). That is we determine to what extent the multifractal
spectrum depends continuously on the cIFS and its thermodynamic potential. For
this we introduce the notion of regular convergence for families of cIFS not
necessarily sharing the same index set, which guarantees the convergence of the
multifractal spectra on the interior of their domain. In particular, we obtain
an Exhausting Principle for infinite cIFS allowing us to carry over results for
finite to infinite systems, and in this way to establish a multifractal
analysis without the usual regularity conditions. Finally, we discuss the
connections to the -topology introduced by Roy and Urbas{\'n}ki.Comment: 16 pages; 3 figure
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