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On the spectra of randomly perturbed expanding maps. (English) Zbl 0809.60101

The authors investigate the relations of exponentially mixing (exponential decay of correlations) and stochastic stability. The considered models are (i) expanding maps of a circle with special perturbations by convolutions, (ii) mixing piecewise expanding maps of the interval with similar perturbation, (iii) expanding maps of Riemannian manifolds perturbed by time-\(\varepsilon\) maps of stochastic flows. For these maps they consider the Perron-Frobenius operators \(L_ \varepsilon\) and \(L\) for the perturbed and unperturbed maps, respectively. They show that their spectra converges as \(\varepsilon\to 0\) at least in a bounded region of the complex plane what enables the authors to prove stochastic stability and the convergence of the exponential decay rate of the correlation.

MSC:

60K40 Other physical applications of random processes
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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