  For an irreducible non-permutation matrix $A$, the triplet
$({{\mathcal{O}}_A},{{\mathcal{D}}_A},\rho^A)$ for the Cuntz-Krieger algebra
${{\mathcal{O}}_A}$, its canonical maximal abelian $C^*$-subalgebra
${{\mathcal{D}}_A}$, and its gauge action $\rho^A$ is called the Cuntz-Krieger
triplet. We introduce a notion of strong Morita equivalence in the
Cuntz-Krieger triplets, and prove that two Cuntz-Krieger triplets
$({{\mathcal{O}}_A},{{\mathcal{D}}_A},\rho^A)$ and
$({{\mathcal{O}}_B},{{\mathcal{D}}_B},\rho^B)$ are strong Morita equivalent if
and only if $A$ and $B$ are strong shift equivalent.
  We also show that the generalized gauge actions on the stabilized
Cuntz-Krieger algebras are cocycle conjugate if the underlying matrices are
strong shift equivalent. By clarifying K-theoretic behavior of the cocycle
conjugacy, we investigate a relationship between cocycle conjugacy of the gauge
actions on the stabilized Cuntz-Krieger algebras and topological conjugacy of
the underlying topological Markov shifts.
