On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps

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 almost-invariant 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.