69 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
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
H\"older-differentiability of Gibbs distribution functions
In this paper we give non-trivial applications of the thermodynamic formalism
to the theory of distribution functions of Gibbs measures (devil's staircases)
supported on limit sets of finitely generated conformal iterated function
systems in . For a large class of these Gibbs states we determine the
Hausdorff dimension of the set of points at which the distribution function of
these measures is not -H\"older-differentiable. The obtained results
give significant extensions of recent work by Darst, Dekking, Falconer, Li,
Morris, and Xiao. In particular, our results clearly show that the results of
these authors have their natural home within thermodynamic formalism.Comment: 13 pages, 2 figure
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