K L Kouptsov et al 2002 Nonlinearity 15 1795 doi:10.1088/0951-7715/15/6/306
Quadratic rational rotations of the torus and dual lattice maps
K L Kouptsov1, J H Lowenstein1 and F Vivaldi2
Show affiliationsRecommended by L Bunimovich
We develop a general formalism for computer-assisted proofs concerning the orbit structure of certain nonergodic 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 eight 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.
- PACS
- MSC
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37E05 Maps of the interval (piecewise continuous, continuous, smooth)
37E45 Rotation numbers and vectors
- Subjects
- Dates
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Issue 6 (November 2002)
Received 9 April 2002, in final form 9 August 2002
Published 16 September 2002
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Quadratic rational rotations of the torus and dual lattice maps
K L Kouptsov et al 2002 Nonlinearity 15 1795
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