The Dynamics of One-Dimensional Quasi-Affine Maps

We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \lfloor \lambda x+\mu \rfloor$, providing a complete description of the map’s periodic points, and of the limit points of every $x\in \mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$, an explicit formula for the number of 2-cycles possessed by the map, and the $\omega$-limit set of any $x\in \mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, $\{-\mathrm{\infty },\mathrm{\infty }\}$, $\{\mathrm{\infty }\}$, or $\{-\mathrm{\infty }\}$.