An Example of Kakutani Equivalent and Strong Orbit Equivalent Substitution Systems That Are Not Conjugate

We present an example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate. Introduction. The motivation for this example came from [2], in which Dartnell, Durand, and Maass show that a minimal Cantor system and a Sturmian subshift are conjugate if and only if they are Kakutani equivalent and orbit equivalent (or equivalently strong orbit equivalent for Sturmian subshifts). In their paper, they posed the question if this is true for general minimal Cantor systems or even for substitution systems. Kosek, Ormes, and Rudolph [7] answered this question negatively by giving an example of orbit equivalent and Kakutani equivalent substitution systems that are not conjugate. Furthermore, in [7] it is shown that if two minimal Cantor systems are Kakutani equivalent by map that extends to a strong orbit equivalence, then the systems are conjugate. The question that we then considered is if two minimal Cantor systems are Kakutani equivalent and strong orbit equivalent, does this mean that the systems are conjugate? The answer to this question is again answered negatively as the substitution systems in this paper provide a counterexample. Background & Definitions. We begin with a minimal Cantor system, i.e. an ordered pair (X,T ) where X is a Cantor space and T : X → X is a minimal homeomorphism. The minimality of T means that every T -orbit is dense in X , i.e. ∀ x ∈ X , the set {T (x) | n ∈ Z} is dense in X . There are several notions of equivalence in dynamical systems. The strongest of these is conjugacy. Two dynamical systems (X,T ) and (Y, S) are conjugate if there exists a homeomorphism h : X → Y such that h ◦ T = S ◦ h. A weaker notion of equivalence is orbit equivalance. With orbit equivalence, the spaces still must be homeomorphic, but the homeomorphism need only preserve the orbits within each system, i.e. (X,T ) and (Y, S) are orbit equivalent if there exists a homeomorphism h : X → Y and functions n,m : X → Z such that for all x ∈ X , h ◦ T (x) = S ◦ h(x) and h ◦ T(x) = S ◦ h(x). We refer to m and n as the orbit cocycles associated to h. We say that the systems are strong orbit equivalent if the cocycles have at most one point of discontinuity each.