Before revision:  In order to elucidate the relationship between our distance between two dynamic metric spaces and the bottleneck distance between their Rips zigzag barcodes, we exploit recent advances in the stability of zigzag persistence ( due to Botnan and Lesnick ). By providing explicit constructions, we prove that for each integer k\geq 1 there exist pairs of DMSs at finite interleaving distance whose k-th persistent homology barcodes are at infinite barcode distance .
After revision:  In order to elucidate the relationship between our distance between two dynamic metric spaces and the bottleneck distance between their Rips zigzag barcodes, we exploit recent advances in the stability of zigzag persistence ( due to Botnan and Lesnick , and to Bjerkevik .