01-149 Vivaldi F., Vladimirov I

Pseudo-randomness of round-off errors in discretised linear maps on the plane (3207K, postscript) Apr 20, 01

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Abstract. We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle. It was recently shown that for a dense set of parameters, the discretized map can be embedded into an expanding $p$-adic dynamical system, which serves as source of deterministic randomness. These systems can be used to generate infinitely many distinct pseudo-random sequences over a finite alphabet, whose average period is conjectured to grow exponentially with the bit-length of the initial condition (the seed). We study some properties of these symbolic sequences, deriving a central limit theorem for the deviations between the round-off and exact orbits, and obtaining bounds concerning repetitions of words. We also explore some asymptotic problems computationally, verifying, among other things, that the occurrence of words of a given length is consistent with that of an abstract Bernoulli sequence.

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