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Dimension approximation of attractors of graph directed IFSs by self-similar sets

Published online by Cambridge University Press:  31 August 2018

ÁBEL FARKAS
ÁBEL FARKAS*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401Israel. e-mail: thesecondabel@gmail.com
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Abstract

We show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 193 - 207
Copyright
Copyright © Cambridge Philosophical Society 2018 

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