# Oberwolfach Reports

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**Volume 4, Issue 2, 2007, pp. 1027–1072**

**DOI: 10.4171/OWR/2007/19**

Published online: 2008-03-31

Mini-Workshop: Geometric Measure Theoretic Approaches to Potentials on Fractals and Manifolds

Peter Grabner^{[1]}, Douglas Hardin

^{[2]}, Edward B. Saff

^{[3]}and Martina Zähle

^{[4]}(1) Technische Universität Graz, Austria

(2) Vanderbilt University, Nashville, United States

(3) Vanderbilt University, Nashville, USA

(4) Friedrich-Schiller-Universität Jena, Germany

The mini-workshop \emph{Geometric Measure Theoretic Approaches to Potentials on
Fractals and Manifolds}, organised by Peter Grabner (Graz), Douglas Hardin
(Vanderbilt), Edward B. Saff (Vanderbilt), and Martina Z\"ahle (Jena) was held
from April 8 to 14, 2007. The meeting had 17 participants from 6 countries.
The participants had background from different areas such as fractal geometry,
geometric measure theory, stochastic processes, and potential theory. This
diversity gave rise to new interactions among the participants. In order to
initiate these interactions and to put the focus on the main themes of the
workshop, the first two days of the workshop were organised around three
introductory lectures:
\begin{description}
\item[Edward B. Saff] An overview of discrete minimal energy problems on
manifolds
\item[Martina Z\"ahle] Classical potential theory and stochastic processes
\item[Pertti Mattila] Geometric measure theory on fractals.
\end{description}
Potential theory and geometric measure theory have many applications and
interactions with various areas of mathematics. The workshop was focussed
especially on applications in probability theory, fractal geometry, discrete
minimum energy, and harmonic analysis.
Stochastic processes are a classical application of potential theory. Several
talks during the workshop were devoted to this area. \emph{Michael Hinz}
discussed Dirichlet form techniques for the approximation of jump processes on
fractal sets. In particular, he studied the influence of a weight function on
the behaviour of the process. \emph{Yimin Xiao} spoke on recent results on the
behaviour of $\alpha$-stable L\'evy processes. He discussed the connection
between L\'evy processes, energy forms, and the corresponding capacities. He
suggested using these ideas for the study of fractal properties of more general
Markov processes. \emph{Martina Z\"ahle} gave an introductory talk on the
interplay between classical potential theory, stochastic processes, and their
traces on fractals. She gave special emphasis to Riesz and Bessel potentials,
as well as the corresponding function spaces. \emph{Jiaxin Hu} discussed
Dirichlet forms on fractals and their domain. On post-critically finite
(p.~c.~f.) fractals rescalings of finite difference operators are used to
construct Dirichlet forms. Estimates for the effective resistance in terms of
the distance were presented. \emph{Zhenquing Chen} presented a new approach to
the definition of reflecting Brownian motion on compact sets with non-smooth
boundary. This is based on discrete approximation by random walks on finer and
finer grids. It is shown that this definition is equivalent to other less
constructive approaches.
Minimisation of discrete and continuous energies is a classical subject of
potential theory. Recently, relations to questions originating from geometric
measure theory arose, which were one of the motivations for this workshop.
\emph{Edward Saff} gave an introductory talk about recent joint work with
S.~Borodachov and D.~Hardin on discrete minimal energy in the hyper-singular
case. In contrast to classical potential theory for energy integrals, the
discrete energy also exists for Riesz-kernels $\|x-y\|^{-s}$ with $s\geq d$
($d$ being the dimension of the set). Indeed, it has been proved by D.~Hardin
and E.~Saff that the configurations of minimal energy for such values of the
Riesz parameter $s$ are uniformly distributed with respect to normalised
$d$-dimensional Hausdorff measure, as the number of points tends to $\infty$.
\emph{Douglas Hardin} presented the ideas of the proof, which is based on
results for $d$-rectifiable sets, as well as self-similarity considerations.
As $s\to\infty$, the minimal energy problem becomes the classical problem of
best packing. This relates the determination of the asymptotic main term of
minimal energy to packing problems. \emph{Matthew Calef} discussed the
limiting case $s=d$, which can be obtained from classical potential theory by
taking limits $s\to d-$ of suitably renormalised potentials. In this case the
measure of minimal energy equals the normalised Hausdorff measure for compact
subsets of $\R^d$ with positive Lebesgue measure. \emph{Johann Brauchart}
discussed the support of the equilibrium measure on sets of revolution in
$\R^3$ for Riesz-potentials with $0~~1$).
Geometric measure theory and potential theory provide important techniques in
the investigation of properties of fractal sets. \emph{Pertti Mattila}
presented methods and results from the geometric measure theory of fractals.
Special emphasis was given to projection properties of fractal sets, such as
the Hausdorff dimension of the ``generic'' projection of a set of given
dimension. \emph{Daniel Mauldin} discussed constructions of fractal sets based
on (possibly infinite) iterated function systems of conformal maps in $\R^n$.
He described tools and techniques which have been developed to analyse the
properties of the limit set: Hausdorff, packing, Minkowski, or packing
dimensions, as well as the quantisation dimension of Gibbs states and
equilibrium measures of various potentials.
Classical potential theory Riesz, Bessel, and more general kernels and positive
harmonic functions were the subject of three talks. \emph{Volodymyr
Andriyevskyy} gave a talk on positive harmonic functions in $\C\setminus E$
depending on geometric properties of the set $E\subset\R$. He discussed
necessary and sufficient conditions for the existence of two linearly
independent positive harmonic functions. \emph{Natalia Zorii} presented results
on the existence of equilibrium measures on non-compact subsets of $\R^n$.
\emph{Peter Dragnev} discussed the supports of equilibrium on the sphere under
the presence of a Riesz external field. He gave applications of his results to
separation of discrete minimal energy point configurations.
Spectral theory on fractal set, its relation to complex dynamics, and
corresponding zeta functions were the subject of two talks. \emph{Michel
Lapidus} gave an introduction to the theory that he and his coauthors
developed of complex dimensions of fractal sets. Relations to tube formul\ae{}
of self-similar fractals, Minkowski measurability, and spectral asymptotics on
sets with fractal boundary were discussed. \emph{Peter Grabner} presented
results on the analytic continuation of the spectral zeta function on certain
self-similar fractals and the relation to the classical Poincar\'e functional
equation from complex dynamics.~~

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Grabner Peter, Hardin Douglas, Saff Edward, Zähle Martina: Mini-Workshop: Geometric Measure Theoretic Approaches to Potentials on Fractals and Manifolds. *Oberwolfach Rep.* 4 (2007), 1027-1072. doi: 10.4171/OWR/2007/19