33 research outputs found
Renewal theorems for a class of processes with dependent interarrival times and applications in geometry
Renewal theorems are developed for point processes with interarrival times
, where is a stochastic
process with finite state space and is
a H\"older continuous function on a subset .
The theorems developed here unify and generalise the key renewal theorem for
discrete measures and Lalley's renewal theorem for counting measures in
symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The
new renewal theorems allow for direct applications to problems in fractal and
hyperbolic geometry; for instance, results on the Minkowski measurability of
self-conformal sets are deduced. Indeed, these geometric problems motivated the
development of the renewal theorems.Comment: 2 figure
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
Minkowski Content and local Minkowski Content for a class of self-conformal sets
We investigate (local) Minkowski measurability of
images of self-similar sets. We show that (local) Minkowski measurability of a
self-similar set implies (local) Minkowski measurability of its image
and provide an explicit formula for the (local) Minkowski content of in
this case. A counterexample is presented which shows that the converse is not
necessarily true. That is, can be Minkowski measurable although is not.
However, we obtain that an average version of the (local) Minkowski content of
both and always exists and also provide an explicit formula for the
relation between the (local) average Minkowski contents of and .Comment: The final publication is available at http://www.springerlink.co
Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems
We characterise fractal sets arising from conformal iterated function systems and conformal graph directed Markov systems for which the Minkowski content and the fractal curvature measures exist. With this, we generalise studies that have been carried out for invariant sets of iterated function systems consisting of similarities
Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable
A long-standing conjecture of Lapidus claims that under certain conditions,
self-similar fractal sets fail to be Minkowski measurable if and only if they
are of lattice type. The theorem was established for fractal subsets of
by Falconer, Lapidus and v.~Frankenhuijsen, and the forward
direction was shown for fractal subsets of , , by
Gatzouras. Since then, much effort has been made to prove the converse. In this
paper, we prove a partial converse by means of renewal theory. Our proof allows
us to recover several previous results in this regard, but is much shorter and
extends to a more general setting; several technical conditions appearing in
previous versions of this result have now been removed.Comment: 20 pages, 6 figure
Exploring the morphology of mitochondria in breast cancer cells through fractal analysis in fluorescence imaging
Exploring the morphology of mitochondria in breast cancer cells through fractal analysis in fluorescence imaging
Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets
We show that the fractal curvature measures of invariant sets of
one-dimensional conformal iterated function systems satisfying the open set
condition exist, if and only if the associated geometric potential function is
nonlattice. Moreover, in the nonlattice situation we obtain that the Minkowski
content exists and prove that the fractal curvature measures are constant
multiples of the -conformal measure, where denotes the
Minkowski dimension of the invariant set. For the first fractal curvature
measure, this constant factor coincides with the Minkowski content of the
invariant set. In the lattice situation we give sufficient conditions for the
Minkowski content of the invariant set to exist, contrasting the fact that the
Minkowski content of a self-similar lattice fractal never exists. However,
every self-similar set satisfying the open set condition exhibits a Minkowski
measurable diffeomorphic image. Both in the lattice
and nonlattice situation average versions of the fractal curvature measures are
shown to always exist.Comment: 36 page