The behaviour under iteration of a class of spatially-discretised quadratic maps

Abstract

Potentially drastic changes in the dynamical behaviour of iterated maps due to rounding errors in computer arithmetic have motivated the study of spatially-discretised maps, whereby the dynamical behaviour of a map is compared to that of a variant obtained by composing the map with a spatial discretisation operator, such as the floor function. In this paper, we study the dynamical behaviour of the spatially-discretised quadratic map $x\mapsto \left\lfloor \lambda x^2 \right\rfloor$, for all values of $\lambda\in\mathbb{R}$. Specifically, we prove that the map possesses at most three non-zero fixed points, and that every orbit of the map either diverges or becomes eventually constant at one of the existing fixed points.