  We will introduce a notion of normal subshifts. A subshift $(\Lambda,\sigma)$
is said to be normal if it satisfies a certain synchronizing property called
$\lambda$-synchronizing and is infinite as a set. We have lots of purely
infinite simple $C^*$-algebras from normal subshifts including irreducible
infinite sofic shifts, Dyck shifts, $\beta$-shifts, and so on. Eventual
conjugacy of one-sided normal subshifts and topological conjugacy of two-sided
normal subshifts are characterized in terms of the associated $C^*$-algebras
and the associated stabilized $C^*$-algebras with its diagonals and gauge
actions, respectively.
