Random perturbations of chaotic dynamical systems. Stability of the spectrum

Abstract

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue'' means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the SBR invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations. Our main tool in the 1D case is a special technique for ``interchanging'' the map and the perturbation, developed in our previous paper, combined with a compactness argument.