# Problem 151

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On $\Sigma _A^+$, define $x \sim y$ if $\sigma ^n x = \sigma^m y$ for some $n,m$. Find topological invariants for $(\Sigma _A^+, \sim )$.

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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.