Notes on Pointwise Finite-Dimensional $2$-Parameter Persistence Modules

In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module M (with a finite convex isotopy subdivision) is isomorphic to a $2$-parameter persistence module N where the restriction of N to each chamber of the parameter space (R,≤)² is a constant functor. Moreover, we show that the convex isotopy subdivision of M induces a finite encoding of M. Finally, we prove that every indecomposable thin $2$-parameter persistence module is isomorphic to a polytope module.