Journal of Modern Dynamics (JMD)

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

Pages: 477 - 543, Issue 3, July 2007      doi:10.3934/jmd.2007.1.477

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Oliver Díaz-Espinosa - Department of Mathematics & Statistics, McMaster University, Hamilton, ON L9H 1X2, Canada (email)
Rafael de la Llave - Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, United States (email)

Abstract: 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.

Keywords:  period doubling, critical circle maps, renormalization, transfer operators, central limit theorem, effective noise.
Mathematics Subject Classification:  Primary: 37E20, 60F05, 37C30; Secondary: 60B10, 37H99.

Received: August 2006;      Revised: April 2007;      Available Online: April 2007.