Hausdorff, Packing and Capacity Dimensions Page: 56
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56
r "JjL1 lognm (1—lognm)-i—k
(27) log^^Ta^) 'Card({aj >0}) j ,
J=0
then
m~^7.Card(7(k)) =
r ^ _n(klognm - l(k))
(28) Tcard({aj > 0})-(^a.) 1 . ■
j=0
Corollary 7.2. The ^-packing pre—measure, P^(K) , is
m—1
greater than card ({a- > 0}) • ( } a .)""*•.
J J
j=0
Lower Entropy and Capacity Dimension
McMullen [28, p. 8] also proved that the capacity
dimension of K is the same as the lower entropy dimension.
Ve establish her.e the capacity and lower entropy dimensions
are uniform on K and the lower entropy dimension is a—stable
with respect to K.
9|C
For <r € G , let
(29) J(„) = f „,([<>,l]2).
For each ball centered in K, there is some J(<r) inside
Br(x).
Theorem 7.3. The capacity dimension is uniform on K.
Proof. Since is subadditive, then, for any a € G*,
(30) dimc J(<r)nK = dimc K.
By the scaling theorem of chapter 3 for packing pre—measures
(31) , dim^, KflBr(x) = dim^ K
for any x e K and r > 0 ■
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Spear, Donald W. Hausdorff, Packing and Capacity Dimensions, dissertation, August 1989; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc330990/m1/61/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .