Hausdorff and packing measure functions of self-similar sets: continuity and measurability
2008 ◽
Vol 28
(5)
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pp. 1635-1655
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Keyword(s):
The Self
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AbstractLetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition. In this paper we show that the maps(1)and(2)are continuous; here$\dim _{\mathsf {H}}$denotes the Hausdorff dimension, ℋsdenotes thes-dimensional Hausdorff measure and 𝒮sdenotes thes-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.
2011 ◽
Vol 32
(3)
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pp. 1101-1115
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2015 ◽
Vol 36
(5)
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pp. 1534-1556
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1985 ◽
Vol 26
(2)
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pp. 115-120
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1995 ◽
Vol 15
(1)
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pp. 77-97
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2004 ◽
Vol 56
(3)
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pp. 529-552
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Keyword(s):
Keyword(s):
2018 ◽
Vol 167
(01)
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pp. 193-207
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2016 ◽
Vol 160
(3)
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pp. 537-563
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