On the isolated spectrum of the Perron-Frobenius operator

We discuss the existence of large isolated (non-unit) eigenvalues of the Pe rron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or 'resonances') are distributions which approach the invaria nt density (or equilibrium distribution) at a rate slower than that prescri bed by the minimal expansion rate. We consider the transitional behaviour o f 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 posses s such isolated eigenvalues is presented. AMS classification scheme numbers : 37A30(primary), 37E05, 37D20, 47A10, 47A15 (secondary).