-
8/3/2019 Christoph Bandt- Analysis on the octagasket
1/39
Analysis on the octagasket
Warsaw, 5 December 2007
Christoph Bandt
Mathematics, University of Greifswald
[email protected]
-
8/3/2019 Christoph Bandt- Analysis on the octagasket
2/39
Overview
1. Motivation
2. Fractals of finite type
3. Fractal n-gons
4. Analysis on finitely ramified fractals
5. Resistance scaling on new spaces
6. Harmonic structure on octagasket
-
8/3/2019 Christoph Bandt- Analysis on the octagasket
3/39
1.1 Fractal analysis
Classical analysis on fractal spaces.
Started with Alexander+Orbach and
Rammal+Toulouse, J. Physique 1982/83
Many names: Kusuoka, Lindstrom, Barlow,Fukushima, Kigami, Sabot,
Kumagai ...
Various approaches:
Laplace operator, Dirichlet forms
Brownian motion Besov spaces harmonic functions resistance
metric
-
8/3/2019 Christoph Bandt- Analysis on the octagasket
4/39
1.2 Main problem of fractal analysis
Well-developed theory, diverse methods.
But: very few examples,all looking like Sierpinski gasket.
A single point will disconnect the space,
at least locally.
Exception: existence of a Brownian motion
on Sierpinski carpets, proved by Barlow+Bass1990-99 (uniqueness,
parameters, self-similarityremain open)We start analysis on a new
class of spaces.(still under construction)
-
8/3/2019 Christoph Bandt- Analysis on the octagasket
5/39
1.3 Why self-similar sets?
Take harmonic functions: they can be defined
in any metric space by the mean-value prop-
erty:
f(x) =1
(S)
Sf(y) d (y)
where S is a small sphere around x and is the
surface measure.
This makes sense if
there are many balls or ball-like sets with
similar structure (symmetry condition)
there is a concept of surface measure.
Both conditions hold on appropriate self-similar
sets: they form a good testbed for analysis.
-
8/3/2019 Christoph Bandt- Analysis on the octagasket
6/39
2.1 Self-similar sets
Mathematically tractable class of fractals
f1,...,fn contracting similitudes on Rn.
On C, fk
(z
) =kz
+ck
withrk
=|k|
