On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.
By endowing the equivalence relation with a suitable topology, it becomes the continuous orbit equivalence defined in . It is completely classified in ( see  for more general case including reducible case).
The Krieger's dimension group appears in the equivalence relation restricted to n=m.