581 research outputs found
Dimensions and spectral triples for fractals in R^N
Two spectral triples are introduced for a class of fractals in R^n. The
definitions of noncommutative Hausdorff dimension and noncommutative tangential
dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch
functionals considered in math.OA/0202108, are studied for the mentioned
fractals endowed with these spectral triples, showing in many cases their
correspondence with classical objects. In particular, for any limit fractal,
the Hausdorff-Besicovitch functionals do not depend on the generalized limit
procedure.Comment: 24 pages, 4 figures. To appear in Proceedings of the Conference
"Operator Algebras and Mathematical Physics" held in Sinaia, Romania, June
2003, O. Bratteli, R. Longo H. Siedentop Eds., Theta Foundation, Bucares
Isodiametric inequality in Carnot groups
The classical isodiametric inequality in the Euclidean space says that balls
maximize the volume among all sets with a given diameter. We consider in this
paper the case of Carnot groups. We prove that for any Carnot group equipped
with a Haar measure one can find a homogeneous distance for which this fails to
hold. We also consider Carnot-Caratheodory distances and prove that this also
fails for these distances as soon as there are length minimizing curves that
stop to be minimizing in finite time. Next we study some connections with the
comparison between Hausdorff and spherical Hausdorff measures, rectifiability
and the generalized 1/2-Besicovitch conjecture giving in particular some cases
where this conjecture fails.Comment: 14 page
Remarks about the besicovitch covering property in Carnot groups of step 3 and higher
We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in contrast with the case of the Heisenberg groups where such distances satisfy BCP
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