On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Baker, S. (2016) On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure. Monatshefte fur Mathematik, 179 (1). pp. 1-13. ISSN 1436-5081 doi: 10.1007/s00605-015-0755-2

Abstract/Summary

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/46864
Identification Number/DOI	10.1007/s00605-015-0755-2
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Springer
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