Abstract
We present an example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate.
Highlights
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
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) = Sn(x) ◦ h(x) and h ◦ T m(x)(x) = S ◦ h(x)
We say that the systems are strong orbit equivalent if the cocycles have at most one point of discontinuity each
Summary
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. In [7] it is shown that if two minimal Cantor systems are Kakutani equivalent by map that extends to a strong orbit equivalence, the systems are conjugate. The question that we considered is if two minimal Cantor systems are Kakutani equivalent and strong orbit equivalent, does this mean that the systems are conjugate?
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
