 06250 O. DiazEspinosa, R. de la Llave
 {Renormalization of
weak noises of arbitrary shape for onedimensional critical
dynamical systems:
Announcement of results and numerical explorations
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Sep 5, 06

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Abstract. We study the effect of noise on onedimensional critical
dynamical systems (that is, maps with a renormalization theory).
We consider in detail two examples of such dynamical systems:
unimodal maps of the interval at the accumulation of perioddoubling
and smooth
homeomorphisms of the circle with a critical point and
with golden mean rotation number.
We show that, if we scale the space and the time, several properties of
the noise (the cumulants or Wickordered moments) satisfy some
scaling relations.
A consequence of the scaling relations
is that a version of the central limit theorem holds. Irrespective of
the shape of the initial noise, if the bare noise
is weak enough, the effective noise becomes close
to Gaussian in several senses that we can make precise.
We notice that the conclusions are false for maps with
positive Lyapunov exponents.
The method of analysis is close in spirit to the study
of scaling limits in renormalization theory.
We also perform several numerical experiments that confirm the
rigorous results and that suggest several conjectures.
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