The Global Structure of Iterated Function Systems Page: 32
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CHAPTER 3
THE STRUCTURE OF THE SETS OF ALL ATTRACTORS AND
NON-ATTRACTORS
3.1. Defining Sets of Iterated Function Systems and Attractors
In this dissertation, we are concerned about the topological structure and dimen-
sion of the sets of attractors and non-attractors of iterated function systems. To that
end, a few definitions are in order.
Let C be a collection of contraction maps from X into X. By the set IFS(X, C)
we mean the set of all iterated function systems on X consisting of finitely many
contractions all of which belong to C. In this setting, if ) e IFS(X, C), then 4I)
{X; pi i E A} for some finite alphabet A, and we can define the function 4)
/C(X) K/C(X) by
4)(K) = U(K),
ieA
that is 4 is the Hutchinson operator for ). Now define
ATT(X, C): {J e /C(X) :]4 e IFS(X, C) so that 4(J) = J.}
If n e N, then we define IFS(X, C, n) as the set of all iterated function systems on
X which consist of exaclty n contraction maps, all of which belong to C. We define
ATT(X, C, n) in a similar manner as before.
Finally, let 0 < E < s < 1, by the set IFS(X, C, n, E, s) we mean the set of
all iterated function systems on X which consist of exaclty n contraction maps all
belonging to C, and whose contraction factors are uniformly bounded below by E and
above by s. ATT(X, C, n, e, s) is defined in a similar fashion as ATT(X, C).32
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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/m1/38/?rotate=90: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .