Imposing restrictions on the allowed sequences of transformations of a standard IFS can give rise to attractors that are attractors of standard IFS consisting of a (larger) finite set of transformations, or a countably infinite set of transformations, or are not the attractor of any standard IFS. Whichever the case it can be read off from the transition graph of the restrictions. * 257 Fractals 1999.07:257-266. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only. √ 3) log(1/2) . The comparison of these calculations was one of the motivations for our explorations. The Rellick, Edgar, Klapper calculation, incorporating which T i can follow which T j , gives the same result as a Moran equation on a larger Fractals 1999.07:257-266. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only.
We continue the program of Bedient et al.1 by investigating some of the ways of embedding IFS with 1-step memory into IFS with 2-step memory, and 1- and 2-step memory into IFS with 3-step memory. This reveals a hierarchy of attractors of m-step memory IFS as subsets of attractors of n-step memory IFS.
By applying a result from the theory of subshifts of finite type,1 we generalize the result of Frame and Lanski2 to IFS with multistep memory. Specifically, we show that for an IFS [Formula: see text] with m-step memory, there is an IFS with 1-step memory (though in general with many more transformations than [Formula: see text]) having the same attractor as [Formula: see text].
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