  The flow equivalence of sofic shifts is examined using results about the
structure of the corresponding covers. A canonical cover generalising the left
Fischer cover to arbitrary sofic shifts is introduced and used to prove that
the left Krieger cover and the past set cover of a sofic shift can be divided
into natural layers. These results are used to find the range of a flow
invariant and to investigate the ideal structure of the universal C^*-algebras
associated to sofic shifts. The right Fischer covers of sofic beta-shifts are
constructed, and it is proved that the covering maps are always 2 to 1. This is
used to construct the corresponding fiber product covers and to classify these
up to flow equivalence. Additionally, the flow equivalence of renewal systems
is studied, and several partial results are obtained in an attempt to find the
range of the Bowen-Franks invariant over the set of renewal systems of finite
type. In particular, it is shown that the Bowen-Franks group is cyclic for
every member of a class of renewal systems known to attain all entropies
realised by shifts of finite type.
