Renormalization and central limit theorem for critical dynamical systems with weak external noise

We study the effect of weak noise on criticalone-dimensional maps; that is, maps with a renormalizationtheory.   &nbsp We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate ofthis convergence.  &nbsp We analyze in detail maps at the accumulation of period doublingand critical circle maps with golden mean rotation number.Using renormalization group methods,we derive scaling relations for several features of the effective noiseafter long periods. We use these scaling relations toshow that the central limit theorem for weak noise holds in both examples.  &nbsp We note that, for the results presented here, it is essential that the maps have parabolic behavior. They arefalse for hyperbolic orbits.

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