Self-similar and self-affine sets; measure of the intersection of two copies
By Márton Elekes, Tamás Keleti, A. M'ath'e. 2007
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Let K be a self-similar or self-affine set in R^d, let \mu be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\in G we have either \mu(K\cap g(K)) 0 \iff int_K (K\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure \mu has a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which \mu(K\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or $g(K)\supset K$ holds.
Subfields: Food Science, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/10164b29b36a7951360b7616f386dc90d0438aee
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In Mathematical Physics
Citations in the corpus
- Lipschitz equivalence of self-similar sets and hyperbolic boundaries (2012, background): “There have been considerable studies on the intersections of Cantor sets (see [7], [3], [4], [6] and references therein).”