O. Diaz-Espinosa, R. de la Llave
Renormalization and
Central limit theorem for critical dynamical systems
with weak external noise
(646K, PDF)
ABSTRACT. We study of 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 noises 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 times. 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.