- 02-148 Kouptsov K. L., Lowenstein J. H, Vivaldi F.
- Quadratic rational rotations of the torus and dual lattice maps
(2556K, zipped postscript)
Mar 26, 02
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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.