Self-similarity

Abstract

A self-similar set is defined as a compact set which is the union of its images under the members of a collection of contractions, the contractions being indexed by a compact set. Self-similarity is characterized by the consideration of points in the self-similar set as limits associated with certain sequences of contractions. Conditions are given for the occurrence of self-similarity. A self-similar set is also treated as a fixed point in hyperspace, and the continuous variation of self-similar sets is shown. © 1993 Rocky Mountain Mathematics Consortium.