Strongly continuous orbit equivalence of one-sided topological Markov shifts

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 $C^{\ast }$-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.