On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps, Discrete Contin

(Communicated by Carlangelo Liverani) Abstract. Perron-Frobenius operators and their eigendecompositions are in-creasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their correspond-ing eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calcu-lations. 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. 1. Introduction. Perron-Frobenius operator (or transfer operator) methods are increasingly being used to solve problems arising in applications. Perron-Frobenius operators and their spectra have been very successful in the identification of molec