On a class of orbit closures with unique minimal sets (Q1089955)

Let \(S=\{0,1\}\) and \(S^*=\cup_{k\geq 1}S^ k\) i.e. \(S^*\) is the set of all blocks in S. A map \(\theta\) : \({\mathbb{N}}\to S^*\) induces a map \({\bar \theta}\): \({\mathbb{N}}^{{\mathbb{Z}}}\to S^{{\mathbb{Z}}}\) once an origin is chosen. The paper deals with 0-1 sequences arising from the above construction when the image of \(\theta\) has infinitely many blocks. A class of such sequences whose orbit closures contain a unique minimal set is investigated. Necessary and sufficient conditions for unique ergodicity are obtained. It is also shown that the orbit closure is uniquely ergodic iff all its points are quasi-regular.