Orbit equivalence and Kakutani equivalence with Sturmian subshifts (Q2703042)

Let \(X\) be a compact Hausdorff space, \(T:X\to X\) be a homeomorphism. The system \((X,T)\) considered as a dynamical system is minimal if all orbits are dense in \(X\) (equivalently: if the only nonempty invariant set is \(X\)). If \(X\) is a Cantor set then such a system is called a Cantor system. A particular class of Cantor systems is the class of subshifts. The authors recall the definition in the first section of the paper as well as definitions of substitution subshifts and Sturmian subshifts. The main result of the paper is presented as Theorem 1.1, saying that a minimal Cantor system and a Sturmian subshift are topologically conjugate if and only if they are orbit equivalent and Kakutani equivalent. The proof is based on the method of a Bratelli-Vershik representation of Sturmian systems. In two appendices there are: the proof of a matrix proposition needed for the proof of Theorem 1.1 and some computation of the dimension group of a Sturmian subshift.