Domokos G., Szasz D.
Ulam's scheme revisited: digital modeling of chaotic attractors via
micro-perturbations
(279K, uuencoded g-zipped postscript)
ABSTRACT. We consider discretizations $f_N$ of expanding maps $f:I \to I$
in the strict sense: i.e. we assume that the only information
available on the map is a finite set of integers. Using this definition
for computability, we show that by adding a random perturbation
of order $1/N$, the invariant measure corresponding to $f$ can
be approximated and we can also give estimates
of the error term. We prove that the randomized discrete
scheme is equivalent to Ulam's scheme applied to the
polygonal approximation of $f$, thus providing a new
interpretation of Ulam's scheme. We also compare the efficiency
of the randomized iterative scheme to the direct
solution of the $N \times N$ linear system.