99-479 D Bosio, F Vivaldi

Round-off errors and p-adic numbers (254K, Postscript) Dec 15, 99

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Abstract. We explore some connections between round-off errors in linear planar rotations and algebraic number theory. We discretize a map on a lattice in such a way as to retain invertibility, restricting the system parameter (the trace) to rational values with power-prime denominator $p^n$. We show that this system can be embedded into a smooth expansive dynamical system over the $p$-adic integers, consisting of multiplication by a unit composed with a Bernoulli shift. In this representation, the original round-off system corresponds to restriction to a dense subset of the $p$-adic integers. These constructs are based on symbolic dynamics and on the representation of the discrete phase space as a ring of integers in a quadratic number field.

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