  Koopman operator describes evolution of observables in the phase space, which
could be used to extract characteristic dynamical features of a nonlinear
system. Here, we show that it is possible to carry out interesting symbolic
partitions based on properly constructed eigenfunctions of the operator for
chaotic maps. The partition boundaries are the extrema of these eigenfunctions,
the accuracy of which is improved by including more basis functions in the
numerical computation. The validity of this scheme is demonstrated in
well-known 1-d and 2-d maps. It seems no obstacle to extend the computation to
nonlinear systems of high dimensions, which provides a possible way of
dissecting complex dynamics.
