The Global Structure of Iterated Function Systems Page: 47
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0 belong to the above collection of iterated function systems. Hence, our original
attractor and {0} are among the attractors of this class of iterated function systems.
Hence any open set containing our original attractor but not containing {0} would
cut the space of attractors and by the Mazurkiewicz Theorem for m-manifolds, the
inductive dimension of this open set would have to be greater than 2k - 1. There-
fore, for every k, we have found an open set whose boundary has small transfinite
inductive dimension which is greater than 2k - 1, and therefore we may conclude that
trind ATT([0, 1], s) > wvo. Q
4.3. Future Research
In this dissertation we discussed topological properties of sets of attractors and
non-attractors. We also discussed dimensional properties of the sets
ATT([0, 1], S, n)
and
ATT([0, 1], s).
In the future we would like to investigate the same topological and dimensional prop-
erties for infinite iterated function systems.
Remaining in the field of finite iterated function systems, we would like to give
a classification of which countable subsets of R" are elements of ATT(X) and which
are in /C(X) \ ATT(X), and we would like to solve contecture 2.11.
Another issue within finite iterated function systems, we would like address is
CONJECTURE 4.9. If J C R" is countable and has infinite Cantor-Bendixson rank,
then J is not an attractor of any iterated function system.
If this conjecture turns out to be true, then we would also like to solve the following
problem47
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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/m1/53/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .