Floer-type bipersistence modules and rectangle barcodes

In this paper, we show that the pointwise finite-dimensional two-parameter persistence module ${\mathbb{H}\mathbb{F}}_{\ast }^{(\bullet ,\bullet ]}$, defined in terms of interlevel filtered Floer homology, is rectangle-decomposable. This allows for the definition of a barcode ${\mathcal{B}}_{\ast }^{(\bullet ,\bullet ]}$ consisting only of rectangles in ${\mathbb{R}}^2$ associated with ${\mathbb{H}\mathbb{F}}_{\ast }^{(\bullet ,\bullet ]}$. We observe that this rectangle barcode contains information about Usher’s boundary depth and spectral invariants developed by Oh, Schwarz, and Viterbo. Moreover, we establish relevant stability results, particularly concerning the bottleneck distance and Hofer’s distance.