 97639 Michael Blank and Gerhard Keller
 Random perturbations of chaotic dynamical systems. Stability of the spectrum.
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Dec 18, 97

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Abstract. For piecewise expanding onedimensional 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 PerronFrobenius
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~\cite{BK95}, combined with a compactness
argument.
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