Perron-Frobenius operators and their eigendecompositions are increasingly
being used as tools of global analysis for higher dimensional systems.
The numerical computation of large, isolated eigenvalues and their corresponding
eigenfunctions can reveal important persistent structures such as almostinvariant
sets, however, often little can be said rigorously about such calculations.
We attempt to explain some of the numerically observed behaviour
by constructing a hyperbolic map with a Perron-Frobenius operator whose
eigendecomposition is representative of numerical calculations for hyperbolic
systems. We explicitly construct an eigenfunction associated with an isolated
eigenvalue and prove that a special form of Ulam's method well approximates
the isolated spectrum and eigenfunctions of this map.