UKURAN KURVATUR FRAKTAL PADA HIMPUNAN SERUPA DIRI; FRACTAL CURVATURE MEASURES ON SELF–SIMILAR SETS

Ari Puji Prasetiyo, Widodo

2013 | Disertasi | PROGRAM STUDI S2 MATEMATIKA

Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the “classical” concepts of curvature are not directly applicable to fractal sets. In order to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets (e.g. fractals), for which classical geometric characteristic such as curvatures is not available, this notions for their –parallel sets * ( ) + are studied instead, expecting that their limiting behaviour as does provide information about the structure of the initial set . In particular, the limiting behaviour of the total curvatures (or intrinsic volumes) ( ), , are investigated as well as weak limits of the corresponding curvature measures ( ) as . This leads to the notions of fractal curvature and fractal curvature measure, respectively. The well–known Minkowski content appears in this concept as one of the fractal curvatures. For certain classes of self–similar sets, results on the existence of (averaged) fractal curvatures are presented. These limits can be calculated explicitly and are in a certain sense “invariants” of the sets, which may help to distinguish and classify fractals. Based on these results also the fractal curvature measures of these sets are characterized.

Kata Kunci : Ukuran kurvatur; Himpunan