  We will prove that one-sided topological Markov shifts $(X_A,\sigma_A)$ and
$(X_B,\sigma_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are
topologically orbit equivalent if and only if there exists an isomorphism
between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their
commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to
the condition that there exists a homeomorphism from $X_A$ to $X_B$
intertwining their topological full groups. We will also study structure of the
automorphisms of ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.
