Periodicity of certain piecewise affine planar maps

We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $\lambda \in \{\frac{\pm 1\pm \sqrt{5}}{2},\pm \sqrt{2},\pm \sqrt{3}\}$ that all integer sequences $(a_k{)}_{k\in \mathbb{Z}}$ satisfying $0\le a_{k-1}+\lambda a_k+a_{k+1}<1$ are periodic.