  For the class of continuous, measure-preserving automorphisms on compact
metric spaces, a procedure is proposed for constructing a sequence of
finite-dimensional approximations to the associated Koopman operator on a
Hilbert space. These finite-dimensional approximations are obtained from the
so-called "periodic approximation" of the underlying automorphism and take the
form of permutation operators. Results are established on how these
discretizations approximate the Koopman operator spectrally. Specificaly, it is
shown that both the spectral measure and the spectral projectors of these
permutation operators converge weakly to their infinite dimensional
counterparts. Based on this result, a numerical method is derived for computing
the spectra of volume-preserving maps on the unit $m$-torus. The discretized
Koopman operator can be constructed from solving a bipartite matching problem
with $\mathcal{O}(\tilde{n}^{3m/2})$ time-complexity, where $\tilde{n}$ denotes
the gridsize on each dimension. By exploiting the permutation structure of the
discretized Koopman operator, it is further shown that the projections and
density functions are computable in $\mathcal{O}(m \tilde{n}^{m} \log
\tilde{n})$ operations using the FFT algorithm. Our method is illustrated on
several classical examples of automorphisms on the torus that contain either a
discrete, continuous, or a mixed spectra. In addition, the spectral properties
of the Chirikov standard map are examined using our method.
