  Representation of a dynamical system in terms of simplifying modes is a
central premise of reduced order modelling and a primary concern of the
increasingly popular DMD (dynamic mode decomposition) empirical interpretation
of Koopman operator analysis of complex systems. In the spirit of optimal
approximation and reduced order modelling the goal of DMD methods and variants
are to describe the dynamical evolution as a linear evolution in an
appropriately transformed lower rank space, as best as possible.
  That Koopman eigenfunctions follow a linear PDE that is solvable by the
method of characteristics yields several interesting relationships between
geometric and algebraic properties.
  Corresponding to freedom to arbitrarily define functions on a data surface,
for each eigenvalue, there are infinitely many eigenfunctions emanating along
characteristics. We focus on contrasting cardinality and equivalence. In
particular, we introduce an equivalence class, "primary eigenfunctions,"
consisting of those eigenfunctions with identical sets of level sets, that
helps contrast algebraic multiplicity from other geometric aspects.
  Popularly, Koopman methods and notably dynamic mode decomposition (DMD) and
variants, allow data-driven study of how measurable functions evolve along
orbits. As far as we know, there has not been an in depth study regarding the
underlying geometry as related to an efficient representation. We present a
construction that leads to functions on the data surface whose corresponding
eigenfunctions are efficient in a least squares sense. We call this
construction optimal Koopman eigenfunction DMD, (oKEEDMD), and we highlight
with examples.
