# American Institute of Mathematical Sciences

July  2007, 1(3): 477-543. doi: 10.3934/jmd.2007.1.477

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

 1 Department of Mathematics & Statistics, McMaster University, Hamilton, ON L9H 1X2, Canada 2 Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, United States

Received  August 2006 Revised  April 2007 Published  April 2007

We study the effect of weak noise on critical one-dimensional maps; that is, maps with a renormalization theory.
We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.
We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.
We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.
Citation: Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
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