Planar Rotation Sequences and Domain Exchange

In [2] it was conjectured that all integer sequences (ak)k∈Z satisfying 0 ≤ ak−1 + λak + ak+1 < 1 (k ∈ Z) for real λ with |λ| < 2 are periodic. This question arose in the study of shift radix systems. The conjecture is trivially true for λ = −1, 0, 1. A computer assisted proof for λ = 1− √ 5 2 was given by Lowenstein, Hatjispyros and Vivaldi [5], where also the solution for λ = 1+ √ 5 2 is mentioned. A short proof (without use of computers) of the latter case was given by the authors [1]. The proof in [5] is based on a torus map which is described in detail by Kouptsov, Lowenstein and Vivaldi [4] for all quadratic λ corresponding to rational rotations (λ = ±1± √ 5 2 ,± √ 2,± √ 3), by heavy use of computers. Important related work is due to Adler, Kitchens and Tresser [3], Poggiaspalla [6], Vivaldi and Lowenstein [7] and others. We present a survey on a new method similar to the one in [5]. It is based on the study of a piecewise affine torus map and allows proving the conjecture for quadratic λ corresponding to rational rotations and determining all possible period lengths.