The Global Structure of Iterated Function Systems Page: 11
View a full description of this dissertation.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
PROOF. First note that by letting f(x) = x-p for x > 0, we have f'(x) = -x-(P-)
and therefore, by the mean value theorem, ]c e [k, k + 1] so that f(k) - f (k + 1)
pc-(P+). In other words, ]c E [k, k + 1] so that
1 1 p
kv (k + 1) Cp+l'
and also note that
P P P
p <p <
(k + 1)P+l- cP+1 - kP+1"
First we give a lower estimate of the lower box-counting dimension. Let
-k = p /(k + 1)
and note that an interval of length Sk can contain at most one point of the set
{1,1/29, 1/39, . . . ,1/k'}.
Therefore, at least k intervals of length Sk are required to cover F. Therefore
log N(F) > log k
- log Sk log (k+1)P+l
p
log k
(p+1)log(k+1) logp
Letting k - oc, so that k -- 0, yields
1
dimBFP > -
p+1
Now, we give an upper estimate of the upper box-counting dimension. Let ~k
p/k(p+I), and note that k/p intervals of length 5k are required to cover [0, 1/kp], leaving
another k- 1 points in Fp not yet covered. Also note that an interval of length Sk can
contain at most one point of the set {1, 1/29, 1/32,..., 1/(k 1)P}. Thus we have
logN (F,) log( +k 1)
- log 0k+1
p
log(7 + k- 1)
(p + 1) log k log p11
Upcoming Pages
Here’s what’s next.
Search Inside
This dissertation can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Dissertation.
Snyder, Jason Edward. The Global Structure of Iterated Function Systems, dissertation, May 2009; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc9917/m1/17/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .