Before revision:  Given a finite dynamic metric space (DMS ), we construct the zigzag simplicial filtration arising from applying the Rips simplicial complex construction (with a fixed scale parameter) to this finite DMS. Upon passing to 0-th homology with field coefficients, we obtain a zigzag persistence module and, based on standard results, we in turn obtain a persistence diagram or barcode from this zigzag persistence module.
After revision:  Given a finite dynamic metric space (DMS ), we construct the zigzag simplicial filtration arising from applying the Rips simplicial complex construction (with a fixed scale parameter) to this finite DMS. Upon passing to 0-th homology with field coefficients, we obtain a zigzag persistence module and, based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module.