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].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.