  The spread between two lines in rational trigonometry replaces the concept of
angle, allowing the complete specification of many geometrical and dynamical
situations which have traditionally been viewed approximately. This paper
investigates the case of powers of a rational spread rotation, and in
particular, a curious periodicity in the prime power decomposition of the
associated values of the spread polynomials, which are the analogs in rational
trigonometry of the Chebyshev polynomials of the first kind. Rational
trigonometry over finite fields plays a role, together with non-Euclidean
geometries.
