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Periodicity of certain piecewise affine planar maps. (English) Zbl 1209.37018

The authors consider a one-parameter family of invertible second-order recursive integer sequences which represent a planar rotation, forced on a lattice by a round-off process. These sequences have been conjectured to be periodic for all parameter values and all initial conditions. The authors prove this conjecture for eight quadratically irrational parameter values for which the rotational angle is rational. The proof is based on embedding the dynamics in a piecewise isometric map of the two-dimensional torus, which features exact scaling. This result improves earlier results by Lowenstein and coworkers, who established periodicity for the same parameter values, and a set of initial conditions having full density. In addition, the present proof reduces computer-assistance to a minimum, namely to the drawing of some intricate diagrams, essential to clarify the geometry.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37E15 Combinatorial dynamics (types of periodic orbits)
11B83 Special sequences and polynomials
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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