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Ulam's scheme revisited: digital modeling of chaotic attractors via micro-perturbations

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  • 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.
    Mathematics Subject Classification: 37M25, 68W20, 37E05.


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