Vivaldi F., Vladimirov I
Pseudo-randomness of round-off errors in discretised linear maps on the plane
(3207K, postscript)
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.