  We introduce a class of subshifts under the name of "standard one-counter
shifts". The standard one-counter shifts are the Markov coded systems of
certain Markov codes that belong to the family of one-counter languages. We
study topological conjugacy and flow equivalence of standard one-counter
shifts. To subshifts there are associated C*-algebras by their $\lambda$-graph
systems. We describe a class of standard one-counter shifts with the property
that the C*-algebra associated to them is simple, while the C*-algebra that is
associated to their inverse is not. This gives examples of subshifts that are
not flow equivalent to their inverse. For a family of highly structured
standard one-counter shifts we compute the K-groups.
