Taming the Cantor fence

Let C ⊂ R be a compact totally disconnected subset of the real line R and [0, 1] ⊂ R, the closed unit interval. In this paper we prove that all topological embeddings of C × [0, 1] into R2 are tame; that is, there exists an ambient homeomorphism which straightens and makes parallel all arc components. It follows that no positive entropy map of C × [0, 1] (which covers a homeomorphism of C) can be “embedded” into a near homeomorphism of R2.