RANDOM PERTURBATIONS OF CHAOTIC DYNAMICAL-SYSTEMS - STABILITY OF THE SPECTRUM

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbati ons of these maps are close to isolated eigenvalues of the unperturbed system. (Here 'eigenvalue' means eigenvalue of the corresponding Perr on-Frobenius operator acting on the space of functions of bounded vari ation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about t he approximation of the Sinai-Bowen-Ruelle invariant measure of such a map. We provide several simple examples showing that for maps with pe riodic turning points and for general multidimensional smooth hyperbol ic maps isolated eigenvalues are typically unstable under random pertu rbations. Our main tool in the one-dimensional case is a special techn ique for 'interchanging' the map and the perturbation, developed in ou r previous paper (Blank M L and Keller G 1997 Stochastic stability ver sus localization in chaotic dynamical systems Nonlinearity 10 81-107), combined with a compactness argument.