Primitive matrices over polynomial semirings

AbstractAn extension of the definition of primitivity of a real nonnegative matrix to matrices with univariate polynomial entries is presented. Based on a suitably adapted notion of irreducibility an analogue of the classical characterization of real nonnegative primitive matrices by irreducibility and aperiodicity for matrices with univariate polynomial entries is given. In particular, univariate polynomials with nonnegative coefficients which admit a power with strictly positive coefficients are characterized. Moreover, a primitivity criterion based on almost linear periodic matrices over dioids is presented

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