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Representation of a dynamical system in terms of simplifying modes is acentral premise of reduced order modelling and a primary concern of theincreasingly popular DMD (dynamic mode decomposition) empirical interpretationof Koopman operator analysis of complex systems. In the spirit of optimalapproximation and reduced order modelling the goal of DMD methods and variantsare to describe the dynamical evolution as a linear evolution in anappropriately transformed lower rank space, as best as possible.  That Koopman eigenfunctions follow a linear PDE that is solvable by themethod of characteristics yields several interesting relationships betweengeometric and algebraic properties.  Corresponding to freedom to arbitrarily define functions on a data surface,for each eigenvalue, there are infinitely many eigenfunctions emanating alongcharacteristics. We focus on contrasting cardinality and equivalence. Inparticular, we introduce an equivalence class, "primary eigenfunctions,"consisting of those eigenfunctions with identical sets of level sets, thathelps contrast algebraic multiplicity from other geometric aspects.  Popularly, Koopman methods and notably dynamic mode decomposition (DMD) andvariants, allow data-driven study of how measurable functions evolve alongorbits. As far as we know, there has not been an in depth study regarding theunderlying geometry as related to an efficient representation. We present aconstruction that leads to functions on the data surface whose correspondingeigenfunctions are efficient in a least squares sense. We call thisconstruction optimal Koopman eigenfunction DMD, (oKEEDMD), and we highlightwith examples.

Geometric Considerations of a Good Dictionary for Koopman Analysis of  Dynamical Systems: Cardinality, 'Primary Eigenfunction,' and Efficient  Representation