Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fractal curvatures and Minkowski content of self-conformal sets   For self-similar fractals, the Minkowski content and fractal curvature have
been introduced as a suitable limit of the geometric characteristics of its
parallel sets, i.e., of uniformly thin coatings of the fractal. For some
self-conformal sets, the surface and Minkowski contents are known to exist.
Conformal iterated function systems are more flexible models than similarities.
This work unifies and extends such results to general self-conformal sets in
R^d. We prove the rescaled volume, surface area, and curvature of parallel sets
converge in a Cesaro average sense of the limit. Fractal Lipschitz-Killing
curvature-direction measures localize these limits to pairs consisting of a
base point in the fractal and any of its normal directions. There is an
integral formula.
  We assume only the popular Open Set Condition for the first-order geometry,
and remove numerous geometric assumptions. For curvatures, we also assume
regularity of the Euclidean distance function to the fractal if the ambient
dimension exceeds three, and a uniform integrability condition to bound the
curvature on "overlap sets". A limited converse shows the integrability
condition is sharp. We discuss simpler, sufficient conditions.
  The main tools are from ergodic theory. Of independent interest is a
multiplicative ergodic theorem: The distortion, how much an iterate of the
conformal iterated function system deviates from its linearization, converges.

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The categories of this paper are: eess
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg