  We construct "barcodes" for the chain complexes over Novikov rings that arise
in Novikov's Morse theory for closed one-forms and in Floer theory on
not-necessarily-monotone symplectic manifolds. In the case of classical Morse
theory these coincide with the barcodes familiar from persistent homology. Our
barcodes completely characterize the filtered chain homotopy type of the chain
complex; in particular they subsume in a natural way previous filtered
Floer-theoretic invariants such as boundary depth and torsion exponents, and
also reflect information about spectral invariants. We moreover prove a
continuity result which is a natural analogue both of the classical bottleneck
stability theorem in persistent homology and of standard continuity results for
spectral invariants, and we use this to prove a C^0-robustness result for the
fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather
different from the standard methods of persistent homology, is based on a
non-Archimedean singular value decomposition for the boundary operator of the
chain complex.
