The Global Structure of Iterated Function Systems Page: 10
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DEFINITION 1.8. Let X be any bounded set and let N6(X) be the smallest number
of sets of diameter less than or equal to S which can cover X. The lower box-counting
dimension and the upper box-counting dimension are defined respectively as follows:
X l log Nb(X)
dimBX = -lil
6,-0 - logS
log N(X)
dimBX = lim og
-o - log 6
If the lower box-counting dimension agrees with the upper box-counting dimension,
then this common value is called the box-counting dimension of X, and
dimBX l.olog Nbs(X)
dime X = hmr
6- - log e
When calculating the box-counting dimension of a set, it is enough to consider
limits as S tends to 0 through any decreasing sequence ~k such that 8k+1 > Ck for
some constant 0 < c < 1. To see this, note that if 5k+1 < S < 6k, then
log Na(F)< log N6k, (F)
-loga -log Sk
log Nsk l (F)
-log k+1 + log(8k+1/8k)
log Nsk l (F)
-log k+1 +log c'
so that
li log NS(X) . lilog Nsk (F)
him <lim
-o - log k--oo - log ~k
The opposite inequality is trivial; the case of lower limits is handled in a similar way.
PROPOSITION 1.9. For any p > 0 let F, = {1/n"},>i U {0}. Then
1
dimB F=
p9+110
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Snyder, Jason Edward. The Global Structure of Iterated Function Systems, dissertation, May 2009; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc9917/m1/16/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .