Elementary properties of circle map sequences (Q1025759)

The purpose of this paper is to study combinatorial and structural properties of circle map sequences. An embedding procedure is introduced which gives a map \(\Phi: \Omega \rightarrow W:=\{R,L\}^{\mathbb{N}}\) from the hull (closure of the set of translates) to the sequence of embedding operations allowing the study of the structure of \(\Omega\). The set of admissible words is classified in terms of their occurrence, and their frequency is determined. This classification gives an alternative proof of the three-distance theorem. In Appendix 1 some basic properties of the embedding procedure are collected, and in Appendix 2 a combinatorial property of circle map sequences is discussed, which follows from the embedding procedure.