 02148 Kouptsov K. L., Lowenstein J. H, Vivaldi F.
 Quadratic rational rotations of the torus and dual lattice maps
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Mar 26, 02

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Abstract. We develop a general formalism for computedassisted 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 roundedoff planar rotations, we establish the global
periodicity of the latter systems, for a set of orbits of full density.
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