Curvature densities of self-similar sets

2010/09/30 by Jan Rataj, Rataj, Jan, Martina Zähle +1
Mathematics · #28A78 #37A30 #53C65 #FOS: Mathematics #Metric Geometry (math.MG) #Prrimary: 28A80 #Seccondary 28A75 #math.MG #msc:28A75 #msc:28A78 #msc:28A80 #msc:37A30 #msc:53C65

paper · pdf · doi:10.48550/arxiv.1009.6162

17 pages

arxiv created 2010/09/30 · arxiv updated 2010/10/01

Abstract

For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated fractal curvature measures, which are all multiples of the corresponding Hausdorff measures on F, due to its self-similarity. This local approach based on ergodic theory for an associated dynamical system enables us to extend former global curvature results.

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