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 $\lambda\in\{\frac{\pm1\pm\sqrt5}2,\pm\sqrt2,\pm\sqrt3\}$ 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.