  We will introduce a notion of strongly continuous orbit equivalence in
one-sided topological Markov shifts. Strongly continuous orbit equivalence
yields a topological conjugacy between their two-sided topological Markov
shifts $(\bar{X}_A, \bar{\sigma}_A)$ and $(\bar{X}_B, \bar{\sigma}_B)$. We
prove that one-sided topological Markov shifts $(X_A, \sigma_A)$ and $(X_B,
\sigma_B)$ are strongly continuous orbit equivalent if and only if there exists
an isomorphism bewteen the Cuntz-Krieger algebras ${\mathcal{O}}_A$ and
${\mathcal{O}}_B$ preserving their maximal commutative $C^*$-subalgebras
$C(X_A)$ and $C(X_B)$ and giving cocycle conjugate gauge actions. An example of
one-sided topological Markov shifts which are strongly continuous orbit
equivalent but not one-sided topologically conjugate is presented.
