  We present a decomposition of the Koopman operator based on the sparse
structure of the underlying dynamical system, allowing one to consider the
system as a family of subsystems interconnected by a graph. Using the intrinsic
properties of the Koopman operator, we show that eigenfunctions for the
subsystems induce eigenfunctions for the whole system. The use of principal
eigenfunctions allows to reverse this result. Similarly for the adjoint
operator, the Perron-Frobenius operator, invariant measures for the dynamical
system induce invariant measures of the subsystems, while constructing
invariant measures from invariant measures of the subsystems is less
straightforward. We address this question and show that under necessary
compatibility assumptions such an invariant measure exists. Based on these
results we demonstrate that the a-priori knowledge of a decomposition of a
dynamical system allows for a reduction of the computational cost for data
driven approaches on the example of the dynamic mode decomposition.
