Mathematics – Dynamical Systems
Scientific paper
2006-05-18
Acta Appl. Math., 112:91-136, 2010
Mathematics
Dynamical Systems
41 pages, 6 figures, incorporates referee comments and references to new results
Scientific paper
We use the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in \$\mathbb{R}^d$. The resulting power series in $\epsilon$ is a fractal extension of Steiner's classical tube formula for convex bodies $K \ci \bRd$. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer $i=0,1,...,d-1$, just as Steiner's does. However, our formula also contains terms for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in "Fractal Geometry and Complex Dimensions" by the first author and Machiel van Frankenhuijsen.
Lapidus Michel L.
Pearse Erin P. J.
No associations
LandOfFree
Tube formulas and complex dimensions of self-similar tilings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Tube formulas and complex dimensions of self-similar tilings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tube formulas and complex dimensions of self-similar tilings will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-678474