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The subleading eigenvalues and associated eigenfunctions of thePerron-Frobenius operator for 2-dimensional area-preserving maps arenumerically investigated. We closely examine the validity of the so-called Ulammethod, a numerical scheme believed to provide eigenvalues and eigenfunctionsof the Perron-Frobenius operator, both for linear and nonlinear maps on thetorus. For the nonlinear case, the second-largest eigenvalues and theassociated eigenfunctions of the Perron-Frobenius operator are investigated bycalculating the Fokker-Planck operator with sufficiently small diffusivity. Onthe basis of numerical schemes thus established, we find that eigenfunctionsfor the subleading eigenvalues exhibit spatially inhomogeneous patterns,especially showing localization around the region where unstable manifolds aresparsely running. Finally, such spatial patterns of the eigenfunction are shownto be very close to the distribution of the maximal finite-time Lyapunovexponents.

Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps