Porous surfaces
Suppose a fractal set E has the form \(U\setminus \cup_{n=1}C_ n\), where U and the disjoint \(C_ n\) are open domains such that \(clos \cup_{n=1}C_ n=clos U\) and \(c_{n+1}\geq c_ n,\) \(c_ n=diam C_ n.\) Denote \(e=\limsup \log n/(-\log c_ n)\) (which is equal to Borel's logarithmic rarefaction of E and to the Besicovitch-Taylor index) and let \(\Delta\) (E) be the upper Minkowski-Bouligand dimension (capacity, box dimension) of E. For \(E\subset {\mathbb{R}}\) it is known that \(e=\Delta (E)\). The author examines the same question in \({\mathbb{R}}^ 2.\) Theorem 1.1: Suppose \(c^ 2_ n=0(| C_ n|)\) (\(| \cdot |\) denotes Lebesgue measure); \(dist(C_ n,E)=0(c_ n);\) \(| \cup C_ n| =| U|;\) length of \(\partial C_ n=0(c_ n);\) the number of connected components of \(U\setminus clos(\cup_{c_ i\geq c_ n}C_ i)\) is 0(n). Then \(e=\Delta (E).\) Theorem 9.1 generalizes Theorem 1.1, and Section 10 discusses the problem in \({\mathbb{R}}^ p\), \(p\geq 3\), which leads to awkward difficulties. Sections 5 and 7 are devoted to an eclectic choice of applications, and Section 9 to a study of the particular class of self-similar sets.
scientific article; zbMATH DE number 4087954
scientific article