  The subleading eigenvalues and associated eigenfunctions of the
Perron-Frobenius operator for 2-dimensional area-preserving maps are
numerically investigated. We closely examine the validity of the so-called Ulam
method, a numerical scheme believed to provide eigenvalues and eigenfunctions
of the Perron-Frobenius operator, both for linear and nonlinear maps on the
torus. For the nonlinear case, the second-largest eigenvalues and the
associated eigenfunctions of the Perron-Frobenius operator are investigated by
calculating the Fokker-Planck operator with sufficiently small diffusivity. On
the basis of numerical schemes thus established, we find that eigenfunctions
for the subleading eigenvalues exhibit spatially inhomogeneous patterns,
especially showing localization around the region where unstable manifolds are
sparsely running. Finally, such spatial patterns of the eigenfunction are shown
to be very close to the distribution of the maximal finite-time Lyapunov
exponents.
