On the Spectrum of Randomly Perturbed Expanding Maps
نویسنده
چکیده
We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving some simple lemmas about the robustness of the spectra of certain operators. These abstract results are then applied to the Perron-Frobenius operators of the models in question. Introduction Let f : M → M be a dynamical system preserving some natural probability measure μ0 with density ρ0. This paper is motivated by the following question: does exponential mixing imply stochastic stability? Roughly speaking, exponential mixing of (f, μ0) means that, for two observables φ and ψ on M , the correlation between φ ◦ f and ψ decays exponentially fast with n. Stochastic stability means that, if we add a small amount of random noise to f , obtaining at noise level ǫ a Markov process with invariant density ρǫ, then ρǫ tends to ρ0 as ǫ tends to zero. The following heuristic argument suggests an affirmative answer to this question. Consider the Perron-Frobenius operator L associated with f , acting on a suitable class of functions. The exponential mixing property is equivalent to the presence of a gap in the spectrum of L between the eigenvalue equal to unity and the “next largest eigenvalue.” Corresponding to the noisy situation is a noisy Perron-Frobenius operator Lǫ, which should not be too different from L, for small ǫ. By the usual geometric arguments for 1991 Mathematics Subject Classification. 58F11 58F30; 58C25 58F03 58F15 58F19 58G32 60J10. V. Baladi started the present work during a postdoctoral fellowship at IBM, T.J. Watson Center. L.-S. Young is partially supported by the National Science Foundation.
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