Abstract
Based on the results from (Mihail and Miculescu in Math. Rep., Bucur. 11(61)(1):21-32, 2009), where the shift space for an infinite iterated function system (IIFS for short) is defined and the relation between this space and the attractor of the IIFS is described, we give a sufficient condition on a family of nonempty subsets of I, where is an IIFS, in order to have the equality , where A means the attractor of and means the attractor of the sub-iterated function system of . In addition, we prove that given an arbitrary infinite cardinal number , if the attractor of an IIFS is of type (this means that there exists a dense subset of it having the cardinal less than or equal to ), where is a complete metric space, then there exists a sub-iterated function system of , having the property that , such that the attractors of and coincide. MSC:28A80, 54H25.
Highlights
Iterated function systems (IFSs) were conceived in the present form by John Hutchinson [ ] and popularized by Michael Barnsley [ ]
In the present paper, using the results from [ ], especially Theorem . , we present a sufficient condition on a family (Ij)j∈L of nonempty subsets of I, where S = (X,i∈I) is an infinite iterated function system (IIFS), in order to have the equality j∈L AIj = A, where A means the attractor of S and AIj means the attractor of the sub-iterated function system SIj = (X,i∈Ij ) of S
We prove that given an arbitrary infinite cardinal number A, if the attractor of an IIFS S = (X,i∈I) is of type A, where (X, d) is a complete metric space, there exists SJ = (X,i∈J ) a sub-iterated function system of S, having the property that card(J) ≤ A, such that the attractors of S and SJ coincide
Summary
Iterated function systems (IFSs) were conceived in the present form by John Hutchinson [ ] and popularized by Michael Barnsley [ ]. We present a sufficient condition on a family (Ij)j∈L of nonempty subsets of I, where S = (X, (fi)i∈I) is an IIFS, in order to have the equality j∈L AIj = A, where A means the attractor of S and AIj means the attractor of the sub-iterated function system SIj = (X, (fi)i∈Ij ) of S.
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