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

Let be real numbers in and be points in . Consider the collection of maps given by fj(x) =lambda jx + (1-lambda(j)) pj. It is a well known result that there exists a unique nonempty compact set satisfying Each has at least one coding, that is a sequence that satisfies We study the size and complexity of the set of codings of a generic when has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every has a continuum of codings. We also show that almost every has a universal coding. Our work makes no assumptions on the existence of holes in and improves upon existing results when it is assumed contains no holes.