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. Moreover, we 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
–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
nonarchimedean singular value decomposition for the boundary operator of the chain
complex.