Kouptsov K. L., Lowenstein J. H, Vivaldi F.
Quadratic rational rotations of the torus and dual lattice maps
(2556K, zipped postscript)

ABSTRACT.  We develop a general formalism for computed-assisted proofs 
concerning the orbit structure of certain non ergodic piecewise 
affine maps of the torus, whose eigenvalues are roots of unity.
For a specific class of maps, we prove that if the trace
is a quadratic irrational (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 exceptional families. 
The proof is based on exact computations with algebraic 
numbers, where units play the role of scaling parameters.
Exploiting a duality existing between these maps and lattice maps
representing rounded-off planar rotations, we establish the global 
periodicity of the latter systems, for a set of orbits of full density.