The dynamical system generated by the floor function $\lfloor\lambda x\rfloor$

TL;DR

This paper studies the fixed points and long-term behavior of the dynamical system generated by the floor function with a real parameter, revealing how the number of fixed points varies with the parameter and describing possible orbit limits.

Contribution

It characterizes the fixed points of the system for different parameter regions and describes the asymptotic behavior of orbits depending on initial conditions and parameter values.

Findings

Existence of regions with exactly m fixed points for each m

Full set of integers as fixed points only when λ=1

Orbit limits can be fixed points, two-periodic orbits, or diverge to infinity

Abstract

We investigate the dynamical system generated by the function $\lfloor\lambda x\rfloor$ defined on $\mathbb R$ and with a parameter $\lambda\in \mathbb R$. For each given $m\in \mathbb N$ we show that there exists a region of values of $\lambda$, where the function has exactly $m$ fixed points (which are non-negative integers), also there is another region for $\lambda$, where there are exactly $m+1$ fixed points (which are non-positive integers). Moreover the full set $\mathbb Z$ of integer numbers is the set of fixed points iff $\lambda=1$. We show that depending on $\lambda$ and on the initial point $x$ the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) $\pm\infty$.