Local trivializations of suspended minimal Cantor systems and the stable orbit-breaking subalgebra

It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting $C^*$-algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice.