Scaling in piecewise linear maps

A general formalism for computed-assisted proofs for the orbit structure of certain non ergodic piecewise affine maps of the torus is developed. For a specific class of maps, it is proved that if the eigenvalues are roots of unity of degree four over the rationals (the simplest nontrivial case, comprising 8 maps), then the periodic orbits are organized into finitely many renormalizable families, with exponentially increasing period, plus a finite number of finite families. The proof is based on exact computations in algebraic number rings, where units play the role of scaling parameters. These results are applied to the analysis of propagation of round-off errors in planar rotations by certain rational angles, proving periodicity for almost all initial conditions.