  The Koopman operator provides a linear perspective on non-linear dynamics by
focusing on the evolution of observables in an invariant subspace. Observables
of interest are typically linearly reconstructed from the Koopman
eigenfunctions. Despite the broad use of Koopman operators over the past few
years, there exist some misconceptions about the applicability of Koopman
operators to dynamical systems with more than one disjoint invariant sets
(e.g., basins of attractions from isolated fixed points). In this work, we
first provide a simple explanation for the mechanism of linear
reconstruction-based Koopman operators of nonlinear systems with multiple
disjoint invariant sets. Next, we discuss the use of discrete symmetry among
such invariant sets to construct Koopman eigenfunctions in a data efficient
manner. Finally, several numerical examples are provided to illustrate the
benefits of exploiting symmetry for learning the Koopman operator.
