Problem 151
On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.
On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.
Comments
By endowing the equivalence relation with a suitable topology, it becomes the continuous orbit equivalence defined in [2]. It is completely classified in [3]( see [1] for more general case including reducible case). The Krieger’s dimension group appears in the equivalence relation restricted to n=m. [1] T. M. Carlsen, S. Eilers, E. Ortega, and G. Restorff, Flow equivalence and orbit equivalence for shifts of finite type and isomorphisms of their groupoids, arXiv:1610.09945. [2] K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Pacific J. Math. 246(2010), 199–225. [3] K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Kyoto J. Math. 54(2014), 863–878.