19 research outputs found
The Einstein Relation on Metric Measure Spaces
This note is based on F. Burghart's master thesis at Stuttgart university
from July 2018, supervised by Prof. Freiberg.
We review the Einstein relation, which connects the Hausdorff, local walk and
spectral dimensions on a space, in the abstract setting of a metric measure
space equipped with a suitable operator. This requires some twists compared to
the usual definitions from fractal geometry. The main result establishes the
invariance of the three involved notions of fractal dimension under
bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more
generally, how the transport of the analytic and stochastic structure behind
the Einstein relation works. While any homeomorphism suffices for this
transport of structure, non-Lipschitz maps distort the Hausdorff and the local
walk dimension in different ways. To illustrate this, we take a look at
H\"older regular transformations and how they influence the local walk
dimension and prove some partial results concerning the Einstein relation on
graphs of fractional Brownian motions. We conclude by giving a short list of
further questions that may help building a general theory of the Einstein
relation.Comment: 28 pages, 3 figure
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
Exact Computation and Approximation of Stochastic and Analytic Parameters of Generalized Sierpinski Gaskets
The interplay of fractal geometry, analysis and stochastics on the one-parameter sequence of self-similar generalized Sierpinski gaskets is studied. An improved algorithm for the exact computation of mean crossing times through the generating graphs SG(m) of generalized Sierpinski gaskets sg(m) for m up to 37 is presented and numerical approximations up to m = 100 are shown. Moreover, an alternative method for the approximation of the mean crossing times, the walk and the spectral dimensions of these fractal sets based on quasi-random so-called rotor walks is developed, confidence bounds are calculated and numerical results are shown and compared with exact values (if available) and with known asymptotic formula