On the isolated spectrum of the Perron-Frobenius operator

We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or 'resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this 'minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented. AMS classification scheme numbers: 37A30(primary), 37E05, 37D20, 47A10, 47A15 (secondary).