Lipschitzian retracts and curves as invariant sets

Abstract

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ ⁿ _i=1 T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.